nlin.SI — Pith

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5

math.AG 2026-05-22 2 theorems

Integrable observables prove Π-hierarchy equivalences

by Xavier Blot, Danilo Lewański +1 more

Beyond descendants: integrable observables for cohomological field theories

They replace psi classes while keeping integrability, establish Miura links to Dubrovin-Zhang and ramification hierarchies, and give a short

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We introduce the concept of integrable observables and propose them as alternatives to the standard Witten's psi classes (a.k.a. descendants in $2D$ quantum gravity) to be coupled with cohomological field theories and their generalisations. The main property of integrable observables is that they retain the integrability properties. We present three examples of integrable observables. The first two recover the Dubrovin-Zhang and double ramification hierarchies, while revealing new structural features in this framework. The third, a new example, builds on recently established properties of the so-called $\mathbb{\Pi}$-class, extending them and placing this class naturally within the theory of integrable systems. Notably, our integrable observables framework yields a proof that the new $\mathbb{\Pi}$-hierarchies are Miura equivalent both to the Dubrovin-Zhang hierarchies and to the double ramification hierarchies. A new very short proof of Witten's conjecture is also provided.

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Top Pith

5

nlin.SI 2026-05-21 2 theorems

Polynomial Hamiltonians yield meromorphic solutions only for degrees 3,4,5,7

by Marta Dell'Atti, Thomas Kecker

Modified Painlev\'e systems with meromorphic solutions for polynomial Hamiltonians of all degrees

Twelve standard forms are obtained, including new quartic and quintic examples, for use in the Painlevé equivalence problem.

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We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the Painlev\'e property, for which the solutions are single-valued outside a set of fixed singularities. Our systems are equivalent to them in the absence of fixed singularities, and give modified Painlev\'e equations otherwise. Using the geometric approach by computing the Okamoto's spaces of initial conditions for certain Hamiltonian systems with general coefficient functions, we obtain differential constraints on these functions for the systems to have only meromorphic solutions. Guided by the Newton polygon of the Hamiltonian function, we obtain all such systems with polynomial Hamiltonian of degree three, four, five, and seven, up to affine equivalence in the dependent variables, while there are none for degree six or degree higher than seven. We thus obtain a list of 12 standard polynomial Hamiltonians that can serve as reference for the Painlev\'e equivalence problem. This list contains also some new Hamiltonians not previously written down, such as quartic Hamiltonians for Painlev\'e I and II, quartic Hamiltonians for the modified Painlev\'e III and V equations, a quintic Hamiltonian for Painlev\'e IV and quintic and septic Hamiltonians for a modified Painlev\'e VI equation.

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math-ph 2026-07-03

Yang-Baxter plus once-per-period rule yields integrable circuits for any geometry

by Miguel García Fernández, Chiara Paletta +1 more

Open-boundary integrable quantum circuits with different geometries

A mapping from transfer-matrix inhomogeneities produces time-periodic open circuits that stay integrable when each bulk gate is used exactly

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We present a complete classification of integrable Yang-Baxter quantum circuits with open boundary conditions and arbitrary circuit geometries. Starting from the standard transfer-matrix construction with two types of staggered inhomogeneities, we derive a general mapping that determines the arrangement of circuit gates in terms of the inhomogeneities and the system size. We conjecture that time-periodic quantum circuits are integrable whenever the local bulk and boundary gates satisfy the Yang-Baxter equation and the same bulk gate is applied exactly once per period to every nearest-neighbor pair of spins. Our construction also provides an algorithm to detect Yang-Baxter integrability for circuits with arbitrary geometries. Furthermore, we introduce a third type of inhomogeneity, denoted by $\rho$, and demonstrate that the minimum possible circuit depth is four. We show that when these $\rho$-inhomogeneities are placed at the endpoints and in their immediate neighborhood, the resulting boundary gates can be interpreted as single gates acting on multiple sites. Our construction is fully general and applies to regular $R$-matrices, both of difference and non-difference type, together with their associated boundary matrices. As an application, we consider two-qubit gates corresponding to 6- and 8-vertex $R$-matrices of non-difference form satisfying the Yang-Baxter equation, and we construct the associated reflection matrices that generate integrable quantum circuits.

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physics.comp-ph 2026-07-02

Soliton dynamics recovered from scattering data without equations

by Seth Minor, Vanja Dukic +1 more

Learning Effective Soliton Dynamics from Scattering Data

Weak-form identification inside the inverse scattering framework yields low-dimensional models that hold in perturbed regimes.

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The inverse scattering transform (IST) provides the standard theoretical framework for deriving soliton dynamics. Traditionally, such derivations have been of an analytical, rather than data-driven, nature. In this paper, we combine the conceptual framework of the IST with weak-form system identification methods to discover effective soliton dynamics directly from observed scattering data, without assuming prior knowledge of the scattering equations. Our method avoids parameterizing solitary waves via ad hoc curve-fitting by working in the scattering domain, yielding interpretable low-dimensional models that remain valid in perturbed and near-integrable regimes. We demonstrate the performance of the proposed approach on synthetic and experimental data governed by shallow-water equations of Korteweg--de Vries-type and recover models that are consistent with canonical IST theory.

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nlin.SI 2026-07-02

Degenerate Lax pair still yields full KdV conservation hierarchy

by Jimmie Adriazola, Gino Biondini +2 more

Learning Lax Pairs: Revisiting the Classical Paradigm

A spectrally degenerate pair classified as fake generates every conserved quantity via its operator algebra, showing the true-fake split is

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A Lax pair $(L,P)$ is sometimes thought of as a structural certificate, in that the spatial operator $L$ carries the spectral data of an integrable system, and its isospectral evolution under $\partial_t L = [L,P]$ encodes the nonlinear dynamics. Yet, experience shows that the correspondence between equations and Lax pairs is much more nuanced than this picture suggests. Equations can admit Lax pairs that fail to encode the expected integrable structure. This paper probes that anomalous corner of the Lax pair landscape through five case studies (the Euler top, the free Schr\"odinger equation, the inviscid Burgers equation, the shallow water system, and the Korteweg--de Vries equation), each illustrating a different way the link to integrability can be distorted. The approach combines analytical calculations with the Sparse Identification of Lax Operators (SILO) framework, which proved useful throughout, in some cases confirming the textbook pair and in others surfacing alternatives worth understanding on their own terms. The recurring lesson across the five cases is that compatibility underdetermines the Lax representation, so that anomalous pairs are regular features of the landscape rather than pathologies. Notably, we show that a spectrally degenerate Korteweg--de Vries Lax pair, classified as fake by standard criteria, still generates the full conservation hierarchy through its operator algebra, which shows that a blunt dichotomy between true and fake Lax pairs can be too reductive.

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nlin.SI 2026-07-02

mKdV finite-gap amplitudes capped by sum of imaginary parts

by Otis C. Wright

Sharp Upper Bound for Amplitudes of Finite-Gap Solutions of the Modified Korteweg-de Vries Equation

The explicit bound is proved sharp from the invariant polynomial and attained by suitable data.

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A direct proof based on commuting finite-dimensional flows and local polynomial invariants is given for a sharp upper bound on the amplitudes of finite-gap solutions of the modified Korteweg-de Vries (mKdV) equation. The maximal amplitude is the sum of the imaginary parts of the upper-half-plane square roots of the roots of the invariant polynomial of the finite-gap solution of the focusing mKdV equation. An analogous formula is established for a bounded class of solutions of the defocusing mKdV equation. The upper bounds are sharp and are explicitly attained by suitable initial data.

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nlin.SI 2026-07-02

Yangian doubles yield off-shell Bethe vectors

by A. Liashyk, S. Pakuliak +1 more

Yangian Doubles and off-Shell Bethe Vectors

The vectors satisfy properties used to derive recurrence relations and confirm eigenvalues in related ggo-invariant models.

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Off-shell Bethe vectors for a generic $\fg$ invariant integrable model are constructed through the currents of the Yangian doubles of the classical series. These off-shell Bethe vectors are shown to satisfy the defining properties which were used in \cite{LPR-RR} to prove the rectangular recurrence relations and verify the eigenvalue property of the on-shell Bethe vectors in $\ggo$-invariant integrable models.

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hep-th 2026-06-30

Auxiliary deformations of 2D BM model lift to 4D

by Daniele Bielli, Mattia Cesàro

The auxiliary-deformed Breitenlohner-Maison model: duality frames and higher-dimensional origin

Both nu- and mu-frames produce a higher-derivative theory without manifest diffeomorphism invariance after uplift via duality-invariant ansa

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The two-dimensional Breitenlohner-Maison (BM) model is a classically integrable subsector of $D=4$ general relativity endowed with two commuting Killing isometries, obtained via Kaluza-Klein reduction to $D=2$. Integrable deformations of such a theory have recently been constructed via auxiliary fields in the so-called $\nu$-frame. In this work we first extend this point of view by deriving the complementary auxiliary field perspective known as $\mu$-frame, and then explicitly construct the uplift to $D=4$ of both descriptions, relying on an ansatz inspired by duality-invariant Lagrangian formulations of Einstein theory. The resulting four-dimensional deformed model thus obtained is a higher-derivative theory which lacks manifest diffeomorphism invariance in both frames. We comment on possible resolutions of this puzzling feature and on the physical interpretation of the model in $D=4$.

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nlin.SI 2026-06-29

Polarization method yields new conserved integrals for any order

by Noah Gie{ss}ing, Yuri B. Suris

Conserved quantities of discretizations by polarization

Algebraic extension shows the quantities are novel starting at order three and preserve key structures in discrete dynamics.

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Recently, a family of unconventional integrators for higher order ODEs with polynomial vector fields was proposed, based on the polarization of vector fields. The simplest instance is the by now famous Kahan discretization for first order ODEs with quadratic vector fields. All these integrators possess remarkable conservation properties. In particular, for the first and the second order Hamiltonian ODEs, the discretization by polarization possesses an integral of motion and an invariant volume form. In this note, we extend our previously proposed algebraic approach to derivation of these integrals to discretizations of ODEs of an arbitrary order. For all orders $\ge 3$, these integrals are new.

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nlin.SI 2026-06-29

NLS transition asymptotics give plane wave plus t^{-1/3} Painlevé-II correction

by Haibing Zhang, Xianguo Geng +1 more

Tritronqu\'ee Painlev\'e-II asymptotics for the focusing nonlinear Schr\"odinger equation on a modulationally unstable background

Leading term stays a plane wave while the coefficient of the first correction comes from a tritronquée solution of an inhomogeneous Painlevé

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We study the long-time asymptotics of the focusing nonlinear Schr\"odinger equation with nonzero boundary conditions in the transition region. Biondini and Mantzavinos showed that, away from the transition curves, the $(x,t)$-plane decomposes into two constant-amplitude plane-wave regions and a central region described by slowly modulated elliptic oscillations. However, their asymptotic formulae are not uniform near the boundaries separating these regions. The purpose of this paper is to resolve this transition problem. Using a double-scaling nonlinear steepest-descent analysis of the associated Riemann--Hilbert problem, we show that the leading term in the transition region is still a plane wave, while the first nontrivial correction is of order $t^{-1/3} $. The coefficient of this correction is expressed in terms of a distinguished tritronqu\'ee solution of an inhomogeneous Painlev\'e-II equation. This Painlev\'e-II tritronqu\'ee structure is also known to appear in the asymptotic analysis of rogue waves of infinite order.

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nlin.SI 2026-06-29

Nakamura conjecture rewritten using reduced trilinear operators

by Takeshi Fukuyama

Reduced Trilinear Reformulation of the Nakamura Conjecture

Both Hirota bilinear and ordinary derivatives fit inside the (a,b,1) sector of Z3-symmetric operators, enabling a direct method.

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The Tomimatsu--Sato (TS) family, characterized by the rotation parameter $q$ and the TS index $\delta=n,$ provides an important class of exact stationary axisymmetric vacuum solutions of Einstein's equations, whose integrable structure is known to be closely related to the $n$-point Toda molecule hierarchy through the Nakamura Conjecture. However, the set of equations appearing in the Nakamura Conjecture contains not only Hirota bilinear derivatives but also ordinary first-derivative terms, and therefore is not formulated entirely within the conventional bilinear algebra. In this paper we introduce a reduced trilinear formulation based on the reduced sector $(a,b,c)\rightarrow(a,b,1)$ of the $Z_3$-symmetric trilinear Hirota operators. We show that both the Hirota bilinear derivatives and the ordinary derivatives appearing in the Nakamura Conjecture can be rewritten completely within this reduced trilinear framework. Consequently, the set of equations admits a formulation in terms of reduced trilinear operators. We further show that the reduced trilinear formulation naturally inherits a Hirota-type direct method. The conventional bilinear spectral factor $k_i-k_j$ is replaced by the $Z_3$-weighted combinations $k_i+\omega k_j$ and $k_i+\omega^2k_j$, providing a direct-method structure characteristic of the reduced trilinear hierarchy. These results suggest that the Toda-molecule description of the Tomimatsu--Sato hierarchy may be viewed as a reduced sector of a broader trilinear framework, and provide a new perspective on the integrable structure of stationary axisymmetric gravity.

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nlin.SI 2026-06-29

Real Painlevé III solutions switch asymptotics at ϰ=1

by Kenta Miyahara, Maxim L. Yattselev

Transition asymptotics for the real solutions of the sinh-Gordon Painlev\'e III equation

A scaling |p|^2 = 1 + e^{2ϰ x} connects singular and smooth cases with elliptic behavior for ϰ between 0 and 1.

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We consider solutions of the sinh-Gordon Painlev\'e III equation \[ u_{xx} + \frac{1}{x} u_x = \sinh u \] that are real on $(0,\infty)$. They are parametrized by the monodromy parameter $p\in\overline{\mathbb{C}}$, $|p|>1$, and an additional real parameter $s^{\mathbb{R}}$ when $p=\infty$. Our previous joint work with A. Its described the asymptotic behavior of these solutions as $x\to\infty$. Here, we describe the transition as $x, p\to \infty$, $2\Im(p)=-s^{\mathbb R}$, between singular solutions ($|p|<\infty$) and smooth solutions ($p=\infty$). In short, if we parametrize $|p|^2 = 1 + e^{2\varkappa x}$, then the smooth exponential asymptotics of the solutions extends to the region $\varkappa>1$, with a change of the leading order term at $\varkappa=2$; at $\varkappa=1$ the exponential behavior transitions into an elliptic asymptotics, which holds for all $0<\varkappa<1$; as $\varkappa$ decays to zero, elliptic asymptotics degenerates into trigonometric one, which holds for all $p$ fixed.

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nlin.SI 2026-06-29

Perturbations create radiative shelf ahead of mKdV kinks

by Nicholas J. Ossi, Barbara Prinari +2 more

Perturbation theory for kinks of the defocusing modified Korteweg-de Vries equation

Leading-order theory supplies evolution equations for kink parameters and shows generic radiation forms in front of the structure.

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In this work we develop an integrable perturbation theory for the defocusing modified Korteweg-de Vries kink solution based on the squared eigenfunction expansion associated with the underlying Zakharov-Shabat scattering problem. We derive the completeness relation for the squared eigenfunctions appropriate to the kink background, establish the adjoint structure needed to handle perturbations of both the continuous and discrete spectral components, and obtain explicit evolution equations for the perturbed kink parameters at leading order. The study of the first order correction shows that perturbations generically produce a radiative shelf in front of the kink. We also apply our results to certain physically relevant perturbations and show that the predictions are consistent with direct numerical simulations.

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cond-mat.stat-mech 2026-06-26

Rule 201 gives exact finite MPO influence matrices for Floquet-PXP

by Katja Klobas

Exact subsystem dynamics in the deterministic Floquet-PXP model

The deterministic model satisfies a closed set of algebraic conditions, enabling exact multi-time autocorrelation functions.

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The dynamics of local subsystems in a thermodynamically large quantum many-body system can be understood as effectively open as the system produces its own effective bath. The action of this bath can be characterised in terms of the so-called influence matrices. In generic situations, the complexity of these objects grows unfavourably with time, however, there exist solvable cases where influence matrices can be characterised exactly even in the presence of non-trivial interactions. Here we show that Rule 201, a deterministic version of the Floquet-PXP model, is one of these solvable instances. Indeed, it admits influence matrices given by a finite-dimensional matrix-product operator (MPO) that solves a finite set of algebraic conditions. We provide the solution, and characterise multi-time autocorrelation functions.

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math-ph 2026-06-26

Ising model on random surfaces reaches (3,4) critical limit

by Maurice Duits, Nathan Hayford +1 more

The Ising Model Coupled to 2D Gravity: Critical Partition Function

Log-partition-function differential converges to (3,4) τ-function, confirming 1990s matrix-model conjectures

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We prove that the differential of the log of the partition function for the $2$-matrix model with quartic interactions converges in a certain double-scaling regime to the differential of the $\boldsymbol{\tau}$-function for the $(3,4)$ string equation. This confirms the convergence of the critical Ising model on random surfaces to the $(3,4)$ topological minimal model, which was stated in the works of Douglas and Shenker, Br\'{e}zin and Kazakov, and Gross and Migdal. Our analysis is based on a steepest-descent analysis of a Riemann-Hilbert problem associated to a family of biorthogonal polynomials. New features in the matching problem in the construction of local parametrices appear.

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nlin.SI 2026-06-26

Pentagram maps on polyhedra are integrable via operator refactorization

by Bao Wang

Integrable pentagram-type maps on polyhedra via partial difference operators

Bijection to spectral data of partial difference operators yields Lax pairs and Poisson brackets for the maps.

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This paper introduces a family of natural generalizations of the pentagram map from polygons to (twisted) polyhedra and proves their integrability through the partial difference operators. A canonical special case, which corresponds to the discrete Laplace transformation of discrete conjugate nets, is investigated in detail. We first establish a canonical bijection between the projective equivalence classes of these polyhedra in $\mathbb{RP}^3$ and the spectral data of doubly periodic partial difference operators modulo the gauge actions. Furthermore, we prove the complete integrability of these pentagram-type maps by explicitly identifying them with the refactorization maps on the Poisson-Lie group of pseudo partial difference operators. This algebraic identification naturally yields an explicit Lax representation and an $r$-matrix induced Poisson bracket for the geometric dynamics.

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nlin.SI 2026-06-26

Generalized Yang-Baxter equations gain site-dependent symmetries

by Pramod Padmanabhan, Somnath Maity +2 more

Symmetries of the Generalized Yang--Baxter Equations

Extra symmetries for more sites sharply reduce the number of inequivalent solutions and the integrable models they generate.

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The generalized Yang-Baxter equations are multi-site versions of the standard Yang-Baxter equation. When spectral parameters are included, such equations are expected to lead to integrable Hamiltonians with local interactions involving multiple degrees of freedom. In this work we characterize both the continuous and discrete symmetries of these equations required to establish an equivalence class of solutions. We find that the set of such symmetries depend on the number of sites on which the equation is supported. In several cases there are more symmetries than the standard Yang-Baxter equation, thus placing heavy constraints on the number of inequivalent solutions and the associated integrable models.

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nlin.SI 2026-06-25

Asymmetric Ablowitz-Ladik hierarchy gains tri-Hamiltonian structure

by Hanxue Zhang, Qiulan Zhao +1 more

Tri-Hamiltonian structure of an asymmetric generalized Ablowitz-Ladik hierarchy and a Frobenius manifold

The local structure is proved at full dispersion via supervariables and its limit yields a Frobenius manifold.

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We construct a local tri-Hamiltonian structure of the asymmetric (3,1)-type generalized Ablowitz-Ladik (gAL) hierarchy at the full-dispersive level and rigorously prove its validity using the supervariable technique. All central invariants of the corresponding bi-Hamiltonian structures are computed. In addition, we construct a Frobenius manifold M arising from the dispersionless limit of this hierarchy and show that the dispersionless limits of the first flows of the (3,1)-type gAL hierarchy belong to the Principal Hierarchy of M.

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nlin.SI 2026-06-25

Exactly 24 spectral classes cover all integrable two-component systems

by Alexander V. Mikhailov, Vladimir Novikov +1 more

Global Results on the Classification of Two-Component Integrable Evolutionary Systems

Necessary and sufficient conditions derived from spectral invariants classify all odd-order polynomial hierarchies and split them into linea

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We derive necessary and sufficient integrability conditions for two-component polynomial evolutionary systems of odd order in $(1+1)$ dimensions. Integrable systems are members of infinite hierarchies of commuting symmetries, which are characterised by their spectral invariants. We prove that there are precisely $24$ possible spectral classes of integrable hierarchies. As an application, we obtain a complete classification of integrable homogeneous hierarchies whose lowest-order equations are of order 3 and 5. The resulting classification naturally splits into two classes. The C--integrable systems are reduced, by means of differential substitutions, to linear--triangular form, while the S--integrable systems are related, through linear changes of variables and differential substitutions, to canonical Drinfeld--Sokolov KdV-type systems associated with affine Lie algebras of rank two.

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math.CA 2026-06-25

2+1D Dym extension reduces to Painlevé II for exact Stefan solutions

by Colin Rogers, Pablo Amster

A novel 2+1-dimensional extended Dym equation: moving boundary problems solvable via Painlev\'e II symmetry reduction

The symmetry reduction supplies closed-form solutions for a class of moving-boundary problems in the extended equation.

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A novel 2+1-dimensional extension of the solitonic Dym equation is shown to admit a Painlev\'e II symmetry reduction which permits the exact solution of a class of Stefan-type moving boundary problems.

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math-ph 2026-06-24

Symmetry reduces even Painlevé IV hierarchy to Flaschka-Newell Painlevé II

by Mohamad Alameddine, Olivier Marchal

A symmetry reduction of the Painlev\'{e} IV hierarchy to the Flaschka-Newell Painlev\'{e} II hierarchy

The map Ψ(−λ)=σ₁Ψ(λ)σ₁ cuts the isomonodromic problem to an invariant submanifold whose dynamics and Hamiltonians match the target hierarchy

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We study the isomonodromic deformation problem associated with rank-two meromorphic connections on the Riemann sphere having one regular singularity and one irregular singularity of even order at infinity, corresponding to the even Painlev\'{e} IV hierarchy. We show that the symmetry $\Psi(-\lambda)= \sigma_1 \Psi(\lambda) \sigma_1$ defines an invariant submanifold whose induced isomonodromic dynamics coincides with the Flaschka-Newell Painlev\'{e} II hierarchy. Under this identification, the corresponding Lax matrices, Darboux coordinates and Hamiltonian structures can be matched explicitly. In particular, the Hamiltonians of the first members of the Flaschka-Newell hierarchy are recovered from the even Painlev\'{e} IV hierarchy. This provides a geometric interpretation of the Flaschka-Newell hierarchy as a symmetry reduction of an isomonodromic deformation problem, complementing its classical description as a similarity reduction of the modified Korteweg-de Vries hierarchy.

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nlin.SI 2026-06-24

Arithmetic progressions classify Lax hierarchies of difference operators

by Sylvain Carpentier

Arithmetic Supports of Lax Difference Hierarchies

Monic finite-band scalar operators admit infinite local flows exactly when exponents form an arithmetic progression, recovering classical la

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We classify monic finite-band scalar difference operators with independent coefficients admitting infinitely many support-preserving flows. We prove that such operators are completely characterized by an arithmetic condition on their support: the exponents must form an arithmetic progression. Conversely, every arithmetic support gives rise to an infinite hierarchy of local Lax flows. As a consequence, finite-band scalar Lax hierarchies with independent coefficients are classified by three integers (N,p,m), corresponding respectively to the leading order, the common difference of the support, and the number of generators. This framework recovers several classical systems, including the Toda, Volterra, Narita--Itoh--Bogoyavlensky, and Blaszak-Marciniak lattices, while simultaneously producing infinitely many additional examples. In particular, the support (-1,1,m) yields a scalar difference Lax representation of the Beffa-Wang hierarchy, and its Belov-Chaltikian reduction in the case m=2.

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nlin.SI 2026-06-23

Katz extension of disc connection supplies extra symmetry in KdV hierarchy

by Martin T. Luu

The KdV vacuum operator and its Katz extension

The extension, written with Kac coordinates of a Weyl class, identifies the vacuum solution and acts as an additional symmetry.

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We define a connection on the formal disc that can be used to single out the vacuum of the Drinfeld-Sokolov KdV hierarchy associated to a simple complex finite-dimensional Lie algebra. As a connection, it has a canonical Katz extension from the disc to the sphere. We express this Katz extension in terms of the Kac coordinates of a suitable Weyl group conjugacy class. As a consequence, we show that the Katz extension has meaning in the context of the integrable hierarchy: It describes an additional symmetry.

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quant-ph 2026-06-23

Infinite hierarchy yields solvable quantum circuits at every level

by Michael A. Rampp, Suhail A. Rather +1 more

Infinite-Level Hierarchy of Solvable Quantum Circuits

New conditions combined with dual unitarity allow exact analysis of correlations and entanglement across all spacetime.

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Dual-unitary circuits have emerged as a paradigm of exactly solvable yet non-integrable quantum dynamics. Recently, a generalization of dual unitarity attempting to extend the phenomenology of exactly solvable circuits has been introduced through a hierarchy of conditions, with dual unitarity as the first level. However, beyond the second level the proposed generalized dual-unitary hierarchy ceases to be solvable in the whole spacetime. We present an infinite hierarchy of solvability conditions remedying this problem. These new conditions can be combined with the generalized dual-unitary hierarchy to obtain circuits for which correlation functions and entanglement dynamics can be analyzed exactly in the whole spacetime. We show that this novel hierarchy possesses non-trivial solutions at every level. Our results demonstrate that dual unitarity can be systematically extended while preserving solvability, opening up investigations of exactly solvable non-integrable systems with more general properties.

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nlin.SI 2026-06-23

Local bihamiltonian structure for (n,1) rational 2D-Toda reductions

by Haonan Qu, Qiulan Zhao

Bihamiltonian structure of the (n,1)-type rational reductions of the 2D-Toda hierarchy

Direct computation yields compatible operators and an (n+1)-dimensional manifold whose principal hierarchy contains the dispersionless flows

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We derive a local bihamiltonian structure for the rational reduction of the 2D-Toda hierarchy (RR2T) of $(n,1)$-type by direct computations, and construct an $(n+1)$-dimensional semisimple generalized Frobenius manifold with non-flat unity whose Principal Hierarchy contains its dispersionless flows.

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nlin.SI 2026-06-23

Harmonic phases reduce (3+1)D Schrödinger systems exactly to (1+1)D

by O.V. Kaptsov

Exact Harmonic Dimensional Reduction and Conformal Lifting for Multicomponent (3+1) Nonlinear Schr\"odinger Systems

Lifting lemma shows every reduced integrable solution satisfies the full multidimensional equations when potential cancels phase gradient sq

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A harmonic dimensional reduction framework is developed for $(3+1)\mathrm{D}$ systems of coupled nonlinear Schr\"odinger-type equations with stationary transverse trapping potentials. The central result is a lifting lemma: if the transverse phase functions are harmonic and the trapping potential exactly cancels the squared phase gradient, the full $(3+1)\mathrm{D}$ system reduces identically to a closed $(1+1)\mathrm{D}$ integrable hierarchy, and every solution of the reduced system lifts to an exact solution of the original multidimensional model. The framework is applied to four systems. For the scalar Gross--Pitaevskii equation, Kuznetsov--Ma breathers are embedded in $(3+1)\mathrm{D}$ geometries carrying vortex lattices with finite, non-singular density at the cores. For the two-component Manakov system, the phase-inversion ansatz yields exact vector solutions with vanishing mass current and non-trivial transverse spin current modulated by the longitudinal breather. For the three-component spinor $F=1$ Bose--Einstein condensate, a symmetric Kuznetsov--Ma breather and a spin-exchange rogue wave are constructed, the latter exhibiting transient density amplification by a factor of nine in the $m_F=0$ channel. For the Maxwell--Bloch system, self-induced transparency solitons, two-soliton elastic collisions, and Kuznetsov--Ma breathers are lifted to full $(3+1)\mathrm{D}$ geometry, with population inversion remaining transversely uniform despite arbitrary phase winding in the cross-section.

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nlin.SI 2026-06-22

Dark soliton gas asymptotes to N-genus theta function

by Dedi Yan, Xianguo Geng +1 more

Long-time asymptotics of a full arbitrary-genus dark soliton gas for the defocusing nonlinear Schrodinger equation

Error is O(t^{-1}) in pure sectors and O(t^{-1/2}) in mixed sectors away from critical rays.

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We introduce a full arbitrary-genus dark soliton gas for the defocusing nonlinear Schr\"odinger equation with finite-density boundary conditions. Starting from a generalized meromorphic Riemann--Hilbert problem with two alternating residue families on each unit-circle arc, we derive an exact thermodynamic limit whose jump matrix contains two nonzero continuum densities. The limiting Riemann--Hilbert problem is uniquely solvable. In contrast with the half dark-soliton gas, every spectral arc of the full gas carries both oscillatory exponentials. We analyze the resulting problem by the Deift--Zhou nonlinear steepest-descent method on a fixed genus-$N$ spectral curve. The moving point in each mixed sector is a stationary factorization-switching point, not a branch point. The active arc is split into two parts and opened crosswise, while lenses are opened around every remaining arc. After removal of exponentially small lens jumps, the model contour therefore retains all $N$ spectral arcs in every self-similar sector. A quotient-curve zero-counting argument proves strict monotonicity of the characteristic velocity and the global ordering of all endpoint velocities, so the self-similar line is divided into $2N+1$ nonempty sectors. The leading term is an $N$-dimensional Riemann-theta finite-gap solution. The error is $O(t^{-1})$ in the $N+1$ pure sectors and $O(t^{-1/2})$ in the $N$ mixed sectors, uniformly away from the critical rays.

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nlin.SI 2026-06-22

Finite invariants expand integrality check for odd-order discrete equations

by Andrei K Svinin

Invariants for a family of discrete equations with the Laurent property

They supply a wider criterion than the Laurent property alone for when generated sequences remain integers.

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Recently, we have found an infinite family of homogeneous discrete equations of odd order possessing the Laurent property. The first representative of this family is the well-known Somos-5 equation, which under certain conditions generates the integer sequence A006721, which has numerous applications. In this work, we construct a finite set of independent invariants for our equations. We show, through examples, that the presence of these invariants allows us to find a more general criterion for the integrality of sequences compared to what the usual Laurent property provides.

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nlin.SI 2026-06-22

Gram-determinant form yields higher-order Manakov solitons

by K N Santhiya, P S Vinayagam

Exploration of Higher-order Nondegenerate Soliton solutions for Manakov model via Gram-determinant Darboux transformation

Compact representation produces explicit third- and fourth-order solutions whose collisions redistribute energy asymmetrically.

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Higher-order nondegenerate soliton solutions of the integrable Coupled Nonlinear Schrodinger system, known as the Manakov system, are obtained using the Darboux transformation associated with the Gram-determinant approach. Starting from the trivial seed solution, the explicit first and second-order soliton solutions are constructed through the direct iterative Darboux procedure. Owing to the rapid increase in algebraic complexity with successive iterations, the higher-order solutions are reformulated through a compact Gram-determinant representation, thereby establishing a unified and systematic framework for their construction. Using this approach, explicit third- and fourth-order solutions are derived, and the general Nth-order case is presented. The resulting structures exhibit asymmetric localized profiles with multiple intensity peaks arising from distinct spectral parameters. The collision dynamics of higher-order nondegenerate solitons reveal features such as asymmetric energy redistribution and complex interaction behaviour. The present formulation provides a systematic and explicit construction of higher-order nondegenerate solitons, offering new insights into multi-component nonlinear wave dynamics.

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nlin.SI 2026-06-18

Two-component Hunter-Saxton equation gets integrable semi-discretization

by Ayako Hori, Yuta Tanaka +2 more

An integrable semi-discretization of the two-component Hunter-Saxton equation

New bilinear system with pseudo 2-reduction yields Casoratian soliton solutions for the discrete version.

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In this paper, we propose an integrable semi-discretization of the two-component Hunter-Saxton (2-HS) equation, which is obtained as the short-wave limit of the two-component Camassa-Holm (2-CH) equation. We also show that the 2-HS equation can be derived from a new set of bilinear equations, distinct from previously known ones, via a pseudo 2-reduction and a hodograph transformation. Furthermore, we construct the N-soliton solutions of both the continuous and semi-discrete systems in Wronskian and Casoratian forms, respectively.

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nlin.SI 2026-06-16

Clifford algebra yields integrable vector Camassa-Holm system

by Andrew N.W. Hone, Vladimir S. Novikov +1 more

Vector peakon equations and isospectral flows in Clifford algebras

Two-component travelling waves reduce to Liouville integrable Hamiltonian system via reciprocal link to Hirota-Satsuma symmetry.

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Starting from a spectral problem posed in a Clifford algebra with $d$ generators and Euclidean signature, we study an integrable, coupled system of PDEs that can be viewed as a vector perturbation of the Camassa--Holm equation with residual orthogonal symmetry. In the two-component case $d=2$, we show that the travelling wave solutions correspond to a Liouville integrable Hamiltonian system with two degrees of freedom, making use of a reciprocal transformation linking the coupled PDEs to a symmetry of the Hirota--Satsuma system. We also present a symmetry classification of all integrable two-component perturbations of Camassa--Holm, and find that besides the $d=2$ system analyzed here, the coupled 2CH system studied by Olver and Rosenau (as well as by Chen, Liu and Zhang, and Falqui), and equations related to either of those systems by Miura transformations, we also obtain a new system that (to the best of our knowledge) has not been reported previously. For the case of an arbitrary number of components $d$, we additionally investigate the short-pulse (high-frequency) regime, in which the limiting dynamics are governed by a vector-valued Hunter-Saxton type system. Furthermore, we provide a detailed analysis of the corresponding measure-valued (weak) solutions associated with this system.

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hep-th 2026-06-15

Instanton carries non-zero observable magnetic and electric fluxes

by C. A. da Silva, L. A. Ferreira

On the gauge-invariant dynamical charges and densities of the 1-instanton solution

Integral non-Abelian Gauss law shows non-vanishing fluxes at unit scaled radius for both instanton and anti-instanton.

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We study the gauge-invariant dynamically conserved charges, and their corresponding densities, for instanton solutions of Yang-Mills theories in four dimensional Euclidean space, for the gauge group $SU(2)$. Those charges were constructed in [1,2] through the integral equations of Yang-Mills theory, using techniques on generalized loop spaces. We use the integral non-Abelian Gauss law to evaluate the gauge-invariant flux of the magnetic and electric non-Abelian fields through spherical surfaces centered at the origin of the instanton solution. From such a flux, we define gauge-invariant charge densities by considering the charge within an infinitesimal spherical shell of radius $r\equiv\sqrt{x_i \, x^i}/\lambda$, with $\lambda$ being the parameter of the instanton solution, defining its size, and $x_i \, x^i = (x^1)^2 + (x^{2})^2 + (x^{3})^2$. We discuss the issue of the reparameterization invariance of the charges and densities, and show that the magnetic and electric fluxes for the instanton and anti-instanton, at $r=1$ and $x^4 = 0$, $x^4$ being the Euclidean time, are non-zero and observable. Our results give an interesting picture of the internal structure of the instanton, and may be important for the properties of the Yang-Mills $\theta$-vacuum.

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hep-th 2026-06-15

One PDE generates all anti-self-dual Einstein metrics with nonzero Lambda

by Maciej Dunajski, Timothy Moy

Heavenly equations in de Sitter space

The Lipstein-Nagy equation unifies the local description in de Sitter space and reduces to Plebański's heavenly equation when Lambda vanishe

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We demonstrate that all anti-self-dual Einstein metrics with non--zero cosmological constant $\Lambda$ locally arise from solutions of a single second order PDE introduced by Lipstein and Nagy. We show how this equation fits into the hyper--heavenly formalism of Pleba\'nski, and establish a Lax pair. Finally we show how Pleba\'nski's second heavenly equation arises in the limit as $\Lambda\rightarrow 0$.

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nlin.SI 2026-06-12

Volterra lattice models A2 Painlevé IV orbits for explicit solutions

by Y. F. Adans, H. Aratyn +2 more

Volterra-Bogoyavlensky lattices and solutions of A_(2n)⁽¹⁾ invariant Painlev\'e equations

Suitable initial conditions produce closed-form expressions in generalized Okamoto polynomials and new recurrence relations.

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The objective of this work is to develop a framework that exploits the lattice structure of the $k$-th Volterra--Bogoyavlensky equations ($k\in\mathbb N$, $k>1$) to generate rational solutions of higher symmetric Painlev\'e equations. For $k=2$, we show that the Volterra lattice, equipped with suitable initial conditions, exactly models the one- and two-dimensional orbits generated by half-translation operators of the $A_2^{(1)}$ symmetric Painlev\'e IV equations. This correspondence yields explicit closed-form expressions for all solution components in terms of generalized Okamoto polynomials and leads to new algebraic recurrence relations among these polynomials. We present two generalizations of the above Volterra lattice. One is derived from a fractional translation of the $A_{4}^{(1)}$ symmetric Painlev\'e equations. It generalizes Volterra lattice structure in the multi-compneent setup of the affine $A_{4}^{(1)}$ group and it is shown to generate solutions of the $A_{4}^{(1)}$ symmetric Painlev\'e equations from the seed solutions invariant under dihedral group $D_{5}$. The other is the $k=3$ Bogoyavlensky lattice structure. It satisfies recurrence relations that naturally extend recurrence relations of the Volterra lattice.

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hep-th 2026-06-12

Only undeformed Maxwell stays causal under relevant deformations

by H. Babaei-Aghbolagh, Bin Chen +2 more

Exact Relevant Stress-Tensor Flows and a Causality No-Go in Self-Dual Electrodynamics

Power-law family of self-dual electrodynamics yields acausality for every nonzero relevant branch.

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Can a classically relevant stress-tensor deformation be exactly solvable, duality preserving, and physically causal? We construct an exact power-law family of nonlinear electrodynamics preserving electromagnetic duality, together with a parallel two-dimensional Lax-integrable realization. Its auxiliary geometry yields the full characteristic-cone phase diagram and a universal finite-energy fold. For the Maxwell seed, every nonzero relevant branch is acausal, whereas every causal branch is caustic-free; undeformed Maxwell theory is the only causal point in the relevant regime.

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nlin.SI 2026-06-12

Quivers yield explicit soliton formulas for KP and Toda systems

by Takashi Ichikawa

Calculation of solitons derived from quivers

An explicit computable formula arises by degenerating quasi-periodic solutions for each quiver.

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We associate each quiver with soliton solutions of nonlinear integrable systems containing the KP and Toda hierarchies. We give an explicit and computable formula of these soliton solutions which are regarded as universal ones obtained by degenerating quasi-periodic solutions.

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nlin.SI 2026-06-10

Camassa-Holm soliton gas limit yields two-reflection RH problem

by Dedi Yan, Xianguo Geng +1 more

Long-time Asymptotics of a Full Camassa-Holm Soliton Gas

Discrete pure-soliton problems converge to a limiting model solved by steepest descent that produces finite-gap elliptic leading terms in th

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We investigate the long-time asymptotics of a full soliton gas for the Camassa--Holm equation. The analysis starts from a pure-soliton Riemann--Hilbert (RH) problem with $2N$ poles and two distinct types of residue conditions. We prove that, as $N\to\infty$, this discrete RH problem converges to a limiting soliton gas RH problem whose jump matrix contains two nonzero reflection coefficients. In this sense, the limiting problem gives a full soliton gas model for the Camassa--Holm equation, in contrast to the previously studied half soliton gas models, whose jump matrices involve only one nonzero reflection coefficient. The limiting RH problem is analyzed by the Deift--Zhou nonlinear steepest descent method. The presence of two nonzero reflection coefficients requires two different types of triangular factorizations of the jump matrix and leads to a more delicate $g$-function mechanism. The main difficulty lies in the construction of suitable $g$-functions adapted to the Camassa--Holm phase, together with the precise control of their behavior near the distinguished point $k=i/2$ and at infinity. Depending on the location of the spectral endpoints $\eta_1$ and $\eta_2$, different $g$-function mechanisms arise. In this paper, we focus on Case I and derive the long-time asymptotic formulas in three elliptic-wave regions of the self-similar plane. In each region, the leading term is given by a finite-gap elliptic function, while in the central region the first correction is of order $\mathcal O(t^{-1/2})$ and involves parabolic cylinder functions.

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nlin.SI 2026-06-10

KdV soliton gas asymptotics given by Jacobi elliptic functions

by Dedi Yan, Xianguo Geng +1 more

Large-time asymptotics of a new KdV soliton gas

A limiting Riemann-Hilbert problem carrying two nonzero reflection coefficients reduces to hyperelliptic model problems whose solutions are

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We study the large-time asymptotic behavior of a new KdV soliton gas. We first introduce a pure-soliton Riemann--Hilbert(RH) problem with $2N$ poles and two different types of residue conditions. We show that, as $N\to\infty$, this discrete problem converges to primitive-potential RH problem introduced by Dyachenko, Zakharov, and Zakharov, and the jump matrix of this soliton gas RH problem has two nonzero reflection coefficients. To analyze the large-time behavior, we apply the Deift--Zhou nonlinear steepest descent method together with an appropriate $g$-function mechanism. Through a sequence of transformations, the original RH problem is reduced to explicitly solvable model problems on an associated hyperelliptic Riemann surface. This allows us to derive an explicit leading-order asymptotic formula for the solution in terms of Jacobi elliptic function. The result provides a rigorous asymptotic description of a new KdV soliton gas and extends the available analysis beyond the previously studied case $r_2\equiv 0$.

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nlin.SI 2026-06-10

Bell polynomials convert sine-Gordon bilinear equations to nonlinear PDEs

by Jin Liu, Da-jun Zhang +2 more

Nonlinearization of bilinear equations of the sine-Gordon type, nonlinear Schr\"odinger type and Benjamin-Ono type

The same conversion works for nonlinear Schrödinger and Benjamin-Ono types, including cases with Hilbert transforms.

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This is a continuation of the paper [Commun. Theor. Phys., 77 (2025) 115006] on the nonlinearization of bilinear equations. The sine-Gordon type and nonlinear Schr\"odinger type bilinear equations are introduced by Jarmo Hietarinta during his search for integrable bilinear equations. In this paper, we provide a formulation to convert these two types of bilinear equations into nonlinear forms. In addition, the nonlinearization related to the equations involving the Hilbert transformations is also considered. Bell polynomials are employed in the nonlinearization and illustrative examples are provided.

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nlin.SI 2026-06-09

Darboux transformations give noncommutative NLS lattice equations

by Sotiris Konstantinou-Rizos

Noncommutative NLS systems: Darboux--B\"acklund transformations and integrable discretisations

Vertex-bond discretisations arise for derivative NLS systems and recover the PDEs in the continuum limit.

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We study noncommutative analogues and integrable discretisations of nonlinear Schr\"odinger (NLS)-type systems associated with reduction groups. In particular, we consider the Ablowitz--Kaup--Newell--Segur (AKNS) system, the Kaup--Newell derivative NLS system, and the Mikhailov--Shabat--Yamilov deformation of the derivative NLS system together with their Darboux--B\"acklund transformations and associated lattice equations. We derive the continuum limits of previously constructed integrable lattice systems and recover the corresponding NLS-type partial differential equations. We then construct a noncommutative deformation of the Mikhailov--Shabat--Yamilov system and show that, unlike the AKNS and Kaup--Newell cases, its Lax representation requires the introduction of nonlocal variables. Furthermore, we derive Darboux--B\"acklund transformations and integrable discretisations for the noncommutative derivative NLS and deformation derivative NLS systems in the form of vertex--bond lattice equations. We also construct explicit solutions for a six-point derivative NLS-type lattice equation and for its noncommutative analogue.

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cond-mat.stat-mech 2026-06-09

Coupled Ising chains give degeneracy-free free-fermion solvability

by Balázs Pozsgay

Free fermions in disguise without exponential degeneracies

A new construction removes the volume-law degeneracies that appeared in all prior free-fermion-in-disguise models.

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Recently, a number of spin chain models have been discovered that are solvable via hidden free-fermionic structures, going beyond the Jordan-Wigner paradigm. However, all examples in the literature displayed degeneracies that grow exponentially with the volume and that are homogeneous in the spectrum (identical degeneracies for all energy levels). In this note we present a model that can be solved by ``free fermions in disguise'' (FFD), such that the spectrum is free from exponential degeneracies for generic coupling constants. The model can be seen as a particular perturbation of two Ising chains. Alternatively, it can be realized as an interpolation between a standard Jordan-Wigner solvable chain and the original FFD model of Fendley. We used ChatGPT Pro 5.4 and 5.5 as a research assistant; in the Supplemental Material we provide details about the collaboration between the AI and the human author.

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nlin.SI 2026-06-08

mKdV waves produce travelling and rotating loops via one integral

by Stephen C. Anco, Jaskaran Maan

Geometric curve flows in the plane and mKdV loop solutions

Each travelling-wave solution of the mKdV equation is turned into an explicit plane curve flow whose loops close only for specific parameter

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There is a well known correspondence between geometric curve flows in the Euclidean plane and solutions of the modified Korteweg-de Vries (mKdV) equation. For each type of mKdV travelling wave, the resulting geometric curve flows are derived here through a simple quadrature formula and studied in detail. These curve flows can be divided into two broad types: travelling loops, and rotating loops. Travelling loops are shown to arise from mKdV solitons, cnoidal (Jacobi cn) and dnoidal (Jacobi dn) waves, the latter being periodic. Rotating loops comprise asymptotically circular ones that are obtained from both mKdV solitary waves on a non-zero background and mKdV rational waves, as well as periodic ones that are produced by mKdV rational elliptic (cn and dn) waves. A specialization of periodic loops, both open and closed, is shown to yield rational cosine loops. An explicit description of each of these types of curve flows is used to characterize their main features, including the condition under which closed loops exist.

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math.AP 2026-06-08

Scattering theory extended to elliptic backgrounds in focusing NLS

by Tamara Grava, Robert Jenkins +2 more

Inverse scattering for the focusing nonlinear Schr\"odinger equation with elliptic background and full soliton gas

Direct and inverse maps exist when bands cross the real axis and the data class overlaps the full soliton gas.

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In this manuscript we develop the direct and inverse scattering problem for the cubic focusing nonlinear Schr\"odinger equation and for initial data that are asymptotic to an elliptic travelling wave with distinct phase at $\pm \infty$. We consider the case in which the spectral bands intersect the real axis. We then show that this class of initial data has non zero intersection with the full soliton gas initial data.

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nlin.SI 2026-06-08

KdV solution obeys no PDE below order one

by Robert Conte (ENS Paris-Saclay, France +2 more

On the Gurevich-Pitaevskii solution of KdV

Gurevich-Pitaevskii solution has converging Laurent series in space and time, so any lower-order equation it satisfies must have differentia

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The universal solution of the Korteweg-de Vries equation (KdV) introduced by Gurevich and Pitaevskii in order to describe the onset of dispersive shock waves is known to also obey the self-similar reduction of the next member in the KdV hierarchy. We show that, if this common solution obeys some lower order partial differential equation, its differential order must be one, and we provide its local representation as a converging Laurent series depending on both space and time.

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math.CO 2026-06-05

Clifford algebra yields recurrences for Grassmannian quantum invariants

by Christian Korff, Mikhail Vasilev

Equivariant Quantum Cohomology of Grassmannians via the Clifford algebra

An explicit Satake map reduces the torus-equivariant quantum cohomology of Grassmannians to projective space and supplies combinatorial posi

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We construct an explicit equivariant quantum Satake map for Grassmannians, which enables us to express their torus-equivariant quantum cohomology in terms of that of projective space. We then consider the exterior algebra of the latter, which admits a canonical identification with a Clifford algebra. We describe the resulting action in several complementary ways: first, from a geometric perspective via push-pull maps, and second, in terms of the shuffle product, which also arises in the simplest cohomological Hall algebra associated with the $A_1$-quiver. Exploiting the Clifford algebra structure, we derive new recurrence relations among equivariant Gromov-Witten invariants, yielding a new method for their computation in terms of Wick's Theorem. As an application, we provide combinatorial proofs of Graham positivity for both equivariant quantum Pieri rules, and in one case extend these results to quantum triple Schubert calculus.

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nlin.SI 2026-06-04

Defocusing Ablowitz-Ladik admits regular time-periodic breathers

by Francesco Coppini, Barbara Prinari

Novel periodic solutions and rogue waves of the defocusing scalar and coupled Ablowitz-Ladik systems on a nonzero background

Choosing the Hirota parameter outside the dark soliton range produces nonsingular solutions, including rogue waves for the coupled system.

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In this paper we apply Hirota's bilinear method to the scalar and coupled Ablowitz-Ladik systems in the defocusing dispersion regime under the assumption of a background amplitude $0<\rho<1$. We first establish, in the scalar case, the correspondence between the Hirota's parameters and the spectral parameters of the inverse scattering transform. Then we show that when the Hirota parameter associated to the discrete eigenvalue is chosen outside the range corresponding to a discrete dark soliton, novel solutions of the Ablowitz-Ladik system emerge. In general, these solutions are singular, but there exists a class of time-periodic solutions for which it is possible to choose the soliton parameters so that the breathers remain regular on the lattice for all times. We also discuss the interactions between a dark soliton and a regular breather, and between two regular breathers. For the coupled Ablowitz-Ladik system, by including in the background discrete, counter-propagating plane waves, we use Hirota's method to derive novel Akhmediev-type (i.e., space-periodic) discrete breathers which are regular for all times. Finally, taking the limit of the discrete Akhmediev breathers as the period approaches infinity (i.e., as the wavenumber approaches zero) we obtain novel rogue wave solutions of the coupled Ablowitz-Ladik system.

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math.DG 2026-06-04

Gl-regular Haantjes operators get complete local description

by Alexey V. Bolsinov, Andrey Yu. Konyaev +1 more

Local description of gl-regular Haantjes operators

The structure of these (1,1)-tensors with zero Haantjes torsion is specified locally, with splitting results and complex eigenvalue cases in

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We study Haantjes operators, that is, (1,1)-tensor fields with vanishing Haantjes torsion. Our main result is a complete local description of gl-regular Haantjes operators. Additional results include a splitting theorem for general (not necessarily gl-regular) Haantjes operators and, more generally, for operators with vanishing generalised Nijenhuis torsion of an arbitrary level, as well as a complete treatment and understanding of the case when the eigenvalues of a Haantjes operator are complex; the latter case was ignored in many previous papers on this and related topics.

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nlin.SI 2026-06-04

Go-diagrams parametrize singular KP solitons from vertex operators

by Shilong Huang

On singular solitons of the KP equation and the Go-diagrams

Products of vertex operators on regular solutions yield singular cases classified by extensions of Le-diagrams in the Grassmannian.

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It has been proven that real and regular soliton solutions of the KP equation are classified in terms of the totally nonnegative Grassmannian. It is well known that vertex operators can be used to construct soliton solutions. In this paper, we consider several regular soliton solutions and study their combinations through products of vertex operators. In general, the resulting solutions become singular. Totally nonnegative elements are parametrized by the Le-diagrams introduced by Postnikov. We show that the resulting singular solutions can be parametrized by Go-diagrams, which extend Le-diagrams and arise in the Deodhar decomposition of the Grassmannian.

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nlin.SI 2026-06-03

Fokas-Lenells equation realized via space-curve curvature and torsion

by R. Ramakrishnan, Sagardeep Talukdar +2 more

Integrable motion of curves associated with the Fokas-Lenells equation and related spin system

Evolution equations match after a complex transformation, extending the geometric link known for simpler integrable systems.

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In this article, we study the gauge equivalence between the integrable Fokas- Lenells equation (FLE) and an associated spin equation through a gauge transformation and the zero curvature condition. We also construct the Lax pair for the generalized spin equation to confirm its integrability. Further, by mapping a generalized spin system on a moving space curve in R3, we show its geometrical equivalence with the FLE. In particular, the associated evolution equations for the curvature and torsion of the space curve are shown to be equivalent to the FLE through a complicated complex transformation unlike the case of the well known Heisenberg spin equation and the nonlinear Schr\"odinger equation.

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nlin.SI 2026-06-03

Heat equation symmetries classified by conductivity function

by I.S. Krasil'shchik

A nonlinear heat transfer equation in turbulent media: symmetry classification, recursion operators, and exact solutions

Group classification by k(T) yields recursion operators in 1D and exact solutions in 1-3 dimensions.

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We study a heat transfer equation in spatial dimensions $n = 1$, $2$, and $3$. A group classification with respect to the functional parameter $k = k(T)$ is done and symmetry algebras are presented. Recursion operators are found in the case $n = 1$ and infinite hierarchies of symmetries are constructed. We also find a number of exact solution in all the three cases.

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math.AP 2026-06-03

Benjamin-Ono solutions follow universal profile from linearized self-similar equation

by Louise Gassot, Patrick Gérard +1 more

The Benjamin-Ono Equation in the Long-Time Limit: Linearized Self-Similar Universality

In the x = O(sqrt(t)) region the leading term decays faster than self-similar solutions for generic rational data.

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We obtain the leading term in the solution of the Cauchy problem for the Benjamin-Ono equation in the limit $t\to+\infty$ with $x=O(t^{1/2})$. We show that the rate of decay exceeds that of self-similar solutions and obtain an explicit universal profile for the decaying solution, relating it to the linearization of the profile equation for self-similar solutions. The proof assumes a class of rational initial data $u_0$ in $L^2(\mathbb{R})\cap L^1(\mathbb{R})$ that exhibit generic behavior of the reflection coefficient at the origin.

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nlin.SI 2026-06-01

Zero-dispersion limits embed algebraic curves in integrable hierarchies

by A. Savchenko, A. Zabrodin

Integrable hierarchies with zero dispersion and elliptic curves

The F-function from the rescaled tau-function limit carries a built-in rational or elliptic dynamical curve for KP, Toda and Pfaff-type syst

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We consider integrable hierarchies such as KP, modified KP, 2D Toda lattice, BKP (small and large), DKP, Pfaff-Toda and their multi-component generalizations. We work in the framework of the bilinear formalism in which the universal dependent variable is a tau-function satisfying bilinear equations of the Hirota-Miwa type. Our principal interest in this paper is the dispersionless versions of the hierarchies. In the limit of zero dispersion the main object is an $F$-function, which is the limit of properly re-scaled logarithm of the tau-function. We show that in all the cases there exists an algebraic curve built into the structure of the hierarchy. We call it the {\it dynamical curve}. For the KP, modified KP and Toda lattice hierarchies, as well as for their multi-component generalizations, the curve is rational (of genus 0) and can be uniformized by rational or trigonometric functions. For hierarchies of the Pfaff type (DKP and Pfaff-Toda) the dynamical curve is in general a smooth elliptic curve (of genus 1), with its modular parameter being a dynamical variable. It is also shown that the large BKP hierarchy admits two different dispersionless versions. In one of them the dynamical curve degenerates to a rational curve while in the other one it remains to be elliptic. We show that a reformulation of the hierarchies based on uniformization of the dynamical curves by elliptic (or trigonometric) functions makes their structure nice and clear, especially in the multi-component case.

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math-ph 2026-06-01

Two first integrals reduce Willmore equation from order three to one

by Z. C. Tu

A first-order reduction of the axisymmetric Willmore equation via two first integrals

The resulting algebraic relation organizes solutions by constants C1 and C2 and recovers spheres and Clifford tori as special cases.

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We derive a first-order formulation of the axisymmetric Willmore equation by combining two independent first integrals, yielding the relation \begin{equation*} \left[\frac{\Psi q^2+2q+2C_1\rho}{\sqrt{1-\Psi^2}}\right]^2 +(q^2-2)^2=C_2, \end{equation*} where $C_1$ and $C_2$ are constants of integration. Here $\rho$ denotes the distance from the axis of revolution, and $q=\rho\Psi'-\Psi$ is the signed hyperbolic curvature of the meridian. We have set $\Psi=\sin\psi$, where $\psi$ is the tangent angle of the meridian, and a prime denotes differentiation with respect to $\rho$. The above relation is equivalent to a quartic algebraic equation for $\rho\Psi'$ and thus reduces the third-order axisymmetric Willmore equation to a first-order ordinary differential equation on each continuous algebraic branch. This formulation recovers the sphere, axisymmetric minimal surfaces, and the Clifford torus as special cases, while organizing the local solution space according to the pair $(C_1,C_2)$. The same hyperbolic formulation also explains how the spontaneous-curvature term destroys the conservation structure underlying the corresponding first integral for axisymmetric Helfrich surfaces.

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math.DS 2026-06-01

C0 and C1 Hamiltonians can be Liouville-Arnold integrable

by Vladimir Dragović, Borislav Gajić +1 more

From bungee to C¹ and C⁰ Hamiltonian systems and their integrability and nonintegrability

Bungee jumping examples show that potentials smooth except on a hypersurface still allow prototype integrable systems.

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We consider natural Hamiltonian systems with potentials that are $C^0$ or $C^1$ on a hypersurface and $C^{\infty}$-smooth in the complement and introduce and study corresponding notions of their integrabilty and non-integrability. As a motivating example, we derive and analyze models of bungee jumping. We provide prototype examples of the Liuoville-Arnol'd theorem for $C^0$ and $C^1$ Hamiltonians.

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cond-mat.stat-mech 2026-06-01

Path products expand hidden free-fermion modes

by Kohei Fukai, Balázs Pozsgay +1 more

Solving models with generalized free fermions II: Path-product expansion and conserved charges

The expansion, obtained from the Krylov basis, also supplies local conserved charges for claw-free graphs and computes dynamical correlation

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Free-fermion solvability in quantum spin systems is increasingly understood to be governed by a graph Clifford algebra defined from the frustration graph of the Hamiltonian. When the frustration graph belongs to certain classes, such as the even-hole-free and claw-free (ECF) class, the Hamiltonian is solvable by hidden free fermions: it admits a free-fermion solution although it does not reduce to a Majorana bilinear under the Jordan-Wigner transformation. However, unlike in the Jordan-Wigner case, where each mode is a linear combination of single Majorana fermions, the explicit operator structure of the hidden free-fermion modes -- and that of the local conserved charges -- has remained obscure. In this work, we derive a path-product expansion that expresses each free-fermion mode as a linear combination of path products along induced paths in the extended frustration graph. The expansion is obtained from the generating function of the Krylov basis and yields the modes directly, without using the transfer matrix or the nonlocal conserved charges as input. As an application, the mode decomposition computes infinite-temperature dynamical correlation functions for arbitrary ECF frustration graphs. We further obtain explicit expressions for local conserved charges as linear combinations of path products along induced paths; these charges apply beyond the free-fermion (ECF) class to more general claw-free frustration graphs. We also identify a unified family of generalized conserved charges that contains both the previously known nonlocal conserved charges and these local conserved charges as special cases. For Fendley's original FFD chain with homogeneous couplings and periodic boundary conditions, in a suitable basis, the structure of these local conserved charges exhibits the same Catalan-tree pattern as in the spin-$1/2$ XXX chain.

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nlin.SI 2026-06-01

Nonlinear source extends symmetries of radial heat equation

by Manjit Singh, Radhika

Lie symmetry classification and group invariant solutions of generalized radial heat equation with nonlinear reaction source

Classification finds extra Lie algebras and exact solutions for specific power and log forms of the constitutive functions.

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This work presents a Lie symmetry classification of a generalized nonlinear heat equation with a reaction source term in radial geometry. The model involves three arbitrary constitutive functions that represent thermal capacity, thermal conductivity, and nonlinear heat generation or absorption. Using the classical Lie invariance criterion, the determining equations for point symmetries are derived and simplified through suitable transformations involving the ratios of the constitutive functions. The classification identifies several admissible subclasses for which the principal symmetry algebra is extended, including power-law and logarithmic branches associated with special values of the radial parameter. For these cases, the admitted Lie algebras, commutator structures, and optimal systems of one-dimensional subalgebras are obtained. The corresponding similarity reductions are constructed, reducing the governing partial differential equation to nonlinear ordinary differential equations. Some exact group-invariant solutions are also derived for special parameter choices. The results show that the inclusion of the nonlinear source term significantly enriches the symmetry structure compared with the source-free radial heat equation.

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cond-mat.stat-mech 2026-05-29

Multi-site spin chain hides Ising free-fermion spectrum with degeneracy

by Akash Sinha, Somnath Maity +2 more

Hidden Ising models from the generalized Yang-Baxter equation

Extra local conserved quantities act as classical background fields, producing multiplicity on top of the transverse-field Ising levels.

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We introduce a one dimensional spin $\frac{1}{2}$ Hamiltonian with multi-site interactions, but still local. The algebra of its Hamiltonian densities resembles that of the transverse field Ising model. Using this fact we show that its spectrum is free-fermionic but with a huge degeneracy for each level. The source of the degeneracy is a set of local conserved quantities that act like a classical background field for the quantum system. The thermodynamics of this system is contrasted with the standard Ising model. At the gapless points in the energy spectrum, we show that this system can be derived from the quantum inverse scattering method adapted to a multi-site generalization of the Yang-Baxter equation as introduced by E. Rowell and Z. Wang. The $R$-matrix is constructed using generators of extraspecial 2-groups. This helps us extract all the conserved charges and lay the framework for a general mechanism to generate such multi-site interaction spin systems that are transverse field Ising models under the hood. A remark on how to obtain P. Fendley's free-fermion in disguise models in this formalism is also included.

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nlin.SI 2026-05-29

Non-commutative discrete Painlevé hierarchy from Lax isomonodromy

by Irina Bobrova

A non-commutative discrete first Painlev\'e hierarchy: the Lax pair approach

The construction gives d-PI_m^nc in Svinin polynomials and reduces from the non-commutative Volterra lattice.

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Using a non-commutative analogue of the isomonodromic problem associated with the discrete first Painlev\'e hierarchy, we construct a non-commutative version of this hierarchy, denoted by $\text{d-PI}_m^{\text{nc}}$. We show that both hierarchies, $\text{d-PI}_m$ and $\text{d-PI}_m^{\text{nc}}$, can be expressed in terms of the polynomials $S_s^k(n)$, which we call the Svinin polynomials. We also derive a reduction of the non-commutative Volterra lattice hierarchy to the $\text{d-PI}_m^{\text{nc}}$ hierarchy and present explicit continuous limits for the first three members of the $\text{d-PI}_m^{\text{nc}}$, thereby recovering non-commutative analogues of the first three members of the differential first Painlev\'e hierarchy.

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hep-th 2026-05-28

Modified boost operator finds new integrable anyonic chains

by Matthew Blakeney, Luke Corcoran +1 more

Constrained integrability and anyonic chains

Extends results to spin-3/2 su(2) models, TY(Z_n) categories, and Fib product categories up to rank 7

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We review the notion of Yang-Baxter integrability for spin chains that have Hilbert spaces with constraints, such as a Rydberg blockade. We focus on anyonic chains, whose constraints arise from the fusion rules of the fusion categories on which they are based. We discuss the emergence of Temperley-Lieb algebras and present a new result on which types of anyonic chains exhibit them. We then give an overview of known results for integrable anyonic chains and extend them to several fusion categories up to rank $7$. Using a modification of the boost operator formalism, we find several new integrable anyonic chains and discuss some of their properties. These include spin-$\frac32$ models for $\mathfrak{su}(2)_k$ fusion categories, anyonic chains based on the Tambara-Yamagami fusion categories TY$(\mathbb{Z}_n)$, and product fusion categories Fib$\times$Fib and Fib$\times$Ising. We review recent results for spin chains based on the Haagerup-Izumi fusion category HI$(\mathbb{Z}_3)$, and present preliminary numerics for a HI$(\mathbb{Z}_5)$ model.

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nlin.SI 2026-05-28

Matrices on sp1(6) yield 1+1 and 2+1 super integrable systems

by Bo Yuan, Yanhui Bi +2 more

Super integrable hierarchies associated with color Lie algebra

Non-isospectral problems on the color Lie algebra produce the systems and the supertrace identity gives their Hamiltonian structures.

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In this paper, by considering two non-isospectral problems with matrices chosen on the color Lie algebra $\mathfrak{sp}_{1}(6)$, we construct (1+1)-dimensional and (2+1)-dimensional super integrable systems on $\mathfrak{sp}_{1}(6)$. Moreover, based on the supertrace identity, their super Hamiltonian structures are also constructed.

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nlin.SI 2026-05-27

Two integrable systems with negative evolutions built in 2+1D

by Andrei Pogrebkov

Integrable equations with negative evolution numbers

Each system includes two evolution equations carrying negative numbers.

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Two integrable systems are constructed in a 2 + 1-dimensional space. Every of these systems involve two evolutions with negative numbers.

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math.AP 2026-05-27

Hybrid ansatz solves moving boundary problems in mKdV system

by Colin Rogers, Adriana C. Briozzo

Hybrid Ermakov-Ray-Reid/Painlev\'e II Symmetry Reduction: Application to a Class of Moving Boundary Problems

Exact solutions arise for a class of nonlinear problems via Ermakov-Ray-Reid and Painlevé II reduction.

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Here, a class of nonlinear moving boundary problems for a novel extension of a two-component mKdV system is shown to admit exact solution via application of a hybrid Ermakov-Ray-Reid / Painlev\'e II symmetry ansatz.The mKdV system has its genesis in a reduction of a coupled nonlinear NLS system incorporating deBroglie - Bohm potential terms.

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nlin.SI 2026-05-25

Generalized Wronskians solve constrained mKP hierarchy

by Jiayi Zhang, Jipeng Cheng +1 more

The generalized Wronskian solutions of the constrained mKP hierarchy

They satisfy the bilinear equations of the (k,m)-constrained modified KP hierarchy and include Wronskian and Grammian cases as special insta

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In this paper, we investigate the $(k, m)$-constrained 1st modified Kadomtsev-Petviashvili (mKP) hierarchy $(L^k)_{\leq 0}= \sum_{i=1}^m q_i \partial^{-1} r_i \partial$. Here, we obtain the corresponding solutions in the form of generalized Wronskians, which include the Wronskians and Grammians as special cases. Most importantly, these generalized Wronskian solutions are proved to satisfy the bilinear equations of the $(k, m)$-constrained mKP hierarchy, which is generally nontrivial. Our results here will be helpful in the derivation of the more general addition formulae and polynomial solutions for the 1st mKP hierarchy.

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hep-th 2026-05-25

Chiral separation on merging branes follows topology-only power law

by H. P. Pinheiro, C. A. S. Almeida

Chirality loss during brane merging: a universal power law from the Jackiw-Rebbi index

Left-right mode distance falls as a power of separation whose exponent depends solely on the Jackiw-Rebbi index, turning Yukawa collapse int

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We investigate the rate at which chiral fermion localisation is lost when two domain walls merge in extra-dimensional braneworld scenarios, using the $(1+1)$-dimensional Jackiw-Rebbi framework as a controlled analytical laboratory. As the inter-brane separation $d$ decreases, left- and right-handed zero modes hybridise and chiral asymmetry is progressively lost. We show that the spatial separation between the chiral zero modes follows a universal power law $|\Delta_{\mathrm{abs}}|\propto d^{\gamma}$ in the merging limit $d\to 0^{+}$, with the critical exponent $\gamma$ determined solely by the Jackiw-Rebbi topological index $N_{\mathrm{JR}}$, and independent of the fermionic mass gap, the integrability of the scalar sector, and the detailed shape of the domain wall profile. Comparing the integrable sine-Gordon model with four members of the non-integrable double sine-Gordon family, all sharing $N_{\mathrm{JR}}=1$, we find $\gamma\in[0.930,0.985]$. For the sine-Gordon model we derive the closed-form overlap integral $I(d)=2d/\sinh(2d)$, from which the exact chiral separation follows as a ratio of hyperbolic functions without free parameters. This result identifies $\gamma$ as the crossover plateau of a local effective exponent $\gamma_{\mathrm{eff}}(d)$, explaining the sub-unit value analytically and tracing the universality to the P\"{o}schl-Teller structure of the $N_{\mathrm{JR}}=1$ zero mode. The universality of $\gamma$ implies that the rate of four-dimensional Yukawa coupling collapse during brane merging is a topological invariant, insensitive to the microscopic scalar dynamics generating the walls.

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nlin.SI 2026-05-25

Hirota method yields explicit soliton-breather solutions for discrete Manakov

by Uyen Le, Alexander Chernyavsky +1 more

Soliton and breather interactions in the integrable discrete focusing Manakov system via Hirota's method

Formulas and asymptotics fully describe all two-body interactions in the focusing regime

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The aim of this paper is to apply Hirota's bilinear method to the integrable discrete Manakov system in the focusing dispersion regime in order to construct and analyze soliton and breather solutions. After deriving the general bilinear form of the system, we show how to obtain fundamental solitons, as well as fundamental and composite breathers. We then obtain solutions exhibiting 2 solitons and 2 breathers and combinations of a soliton and a breather, and discuss all ``two-body'' interactions properties, with particular emphasis on explicit formulas, visualization, and long-time asymptotic behavior, thus rigorously confirming the highly nontrivial interaction properties of these coherent structures.

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nlin.SI 2026-05-22

Cuspon Stefan problems reduce to Painlevé II

by Colin Rogers, Sandra Carillo

Moving Boundary Problems for a Cuspon Equation and Reciprocal Associates: Exact Solution via Painleve' Symmetry Reduction

Symmetry reduction supplies exact solutions for moving-boundary problems on a cuspon equation and its reciprocal soliton associates.

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Here classes of moving boundary problems of Stefan-type for both an established non-linear evolution equation of cuspon theory and novel reciprocally linked solitonic equations are shown to be solvable via Painleve' II symmetry reduction.

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nlin.SI 2026-05-22 2 theorems

Champagne bottle potential yields explicit semi-global invariant

by Ognyan Christov

Semi-global symplectic invariant of the champagne bottle

Birkhoff normal form calculation near the focus-focus point produces the invariant for this monodromy-carrying integrable system.

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We study a two degrees of freedom Hamiltonian system describing the motion of a particle in a potential field of the form of $S^1$ symmetric double well, namely $V = - (x_1^2 + x_2^2) + (x_1^2 + x_2^2)^2$, known also as a champagne bottle potential. This system is completely integrable. The champagne bottle is the simplest member of a class of integrable systems that have no global action variables due to a non-trivial monodromy, Bates (1991). Beyond that, the geometric and dynamical properties of the system near the equilibrium are of primary interest. We calculate the Birkhoff normal form and the nontrivial action near the focus-focus singularity and obtain the semi-global symplectic invariant near focus-focus point, which is introduced by V\~{u} Ng\d{o}c (2003). Examples of such calculations are still few. We compare our result with the semi-global symplectic invariant of the spherical pendulum, calculated by Dullin (2013).

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math-ph 2026-05-21 2 theorems

Prepotentials for type A Frobenius manifolds proven as limits

by Alessandro Proserpio, Karoline van Gemst

Landau-Ginzburg models of generalised Dubrovin-Zhang form and pole collision: Dynkin-type A

Structural result holds for arbitrary rank and proves conjecture on doubly-extended Weyl groups.

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In arXiv:1711.05958, arXiv:2103.12673, the authors derive one-dimensional Landau-Ginzburg mirrors of Dubrovin-Zhang Frobenius manifolds constructed on regular orbit spaces of an extension of affine Weyl groups. We generalise the method employed, and classify the resulting Frobenius manifold structures in Dynkin type A. We interpret our results in terms of a stratification on the Hurwitz space boundary, and develop a pole-collision framework to compare the Frobenius structures within different strata. With this, we can prove a structural result at the level of the prepotential, for arbitrary rank and dimension, as a suitable renormalised limit of the formulae in arXiv:2412.05165. As a corollary, a conjecture of Ma and Zuo regarding the form of prepotentials related to doubly-extended affine Weyl groups is proven.

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nlin.SI 2026-05-21 Recognition

Soliton gas for NLS settles to constant elliptic wave at large negative space

by Dedi Yan, Xianguo Geng +1 more

Large-space and Large-time Asymptotics for the Focusing Nonlinear Schr\"{o}dinger Soliton Gas

Under endpoint assumption, large time splits solution into exponential decay, modulated elliptic waves, and unmodulated elliptic waves.

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We investigate the large-space and large-time asymptotic behavior of a soliton gas for the focusing nonlinear Schr\"odinger equation. The soliton gas is constructed as the continuum limit of pure $N$-soliton solutions as $N\to\infty$, with the discrete spectrum confined to two segments $\Sigma_1$ and $\Sigma_2$. In particular, our framework does not require the discrete spectrum to be confined to the imaginary axis. By combining the nonlinear steepest descent method with an appropriate $g$-function mechanism, we show that, as $x\to-\infty$, the soliton gas is asymptotically described by a finite-gap elliptic solution with constant coefficients. In the large-time regime $t\to+\infty$, we assume that the endpoint $F$ lies on the trajectory of $H(\xi)$ with $\xi=\frac{x}{2t}\in(-E_1-\sqrt{2}E_2,-E_1)$, namely, $F=H(\hat{\xi})$, $\hat{\xi}\in (-E_1-\sqrt{2}E_2,-E_1)$. Under this assumption, we prove that the solution exhibits distinct asymptotic behaviors in different regions of the variable $\xi=\frac{x}{2t}$. More precisely, there exist an exponentially decaying region $\xi\in(-E_1,+\infty)$, a modulated elliptic-wave region $\xi\in(\hat{\xi},-E_1)$, and an unmodulated elliptic-wave region $\xi\in(-\infty,\hat{\xi})$.

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math-ph 2026-05-20 2 theorems

Weakest quantization conditions for deformed Schrödinger equations derived

by Jonah Baerman, Alba Grassi +1 more

Higher-Rank Connections and Deformed Schr\"odinger Operators

Monodromy data formulation proves topological string/spectral theory duality predictions for equations tied to the quantum Toda chain.

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We study the connection problem for a class of linear differential equations of order $N$ closely related to the Baxter equation of the quantum Toda chain. The space of solutions is $N$-dimensional and several linearly independent solutions decay at each singularity, leading to a rich structure of boundary value problems. We derive the weakest quantization conditions compatible with decaying behavior at both singularities, and formulate these conditions in terms of the associated monodromy data. In doing so, we prove the quantization conditions predicted by the topological string/spectral theory duality for a family of deformed Schr\"odinger equations. More generally, our results point to a hierarchy of spectral problems interpolating between the minimal conditions studied here and the maximally decaying boundary conditions of the $N$-particle quantum Toda chain.

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nlin.SI 2026-05-20 1 theorem

mKdV soliton gas splits into 2n+1 theta regions at large time

by Dedi Yan, Xianguo Geng +1 more

Large-space and large-time asymptotics for the mKdV soliton gas with any odd genus

As x tends to positive infinity the profile settles to a single genus-2n-1 theta function; each of the 2n+1 large-t zones carries its own un

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We study the large-space and large-time asymptotic behavior of the soliton gas of genus $2n-1$ for the mKdV equation with $n\in \mathbb{N}_+$. As $x \to +\infty$, we show that the large-space asymptotics of the mKdV soliton gas can be expressed with the Riemann-theta function of genus $2n-1$. For large $t$, based on the nonlinear steepest descent method and $g$-function approach, we establish a global large-time asymptotic description of the mKdV soliton gas. The half-plane $\{(x,t):-\infty<x<+\infty, t>0\}$ is divided into $2n+1$ separated regions. In each region, the large-time asymptotics of the mKdV soliton gas is given by using the Riemann-theta functions and uniform error estimation.

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math.AP 2026-05-20 Recognition

Variational proof establishes stability of smooth waves in Fornberg-Whitham model

by Xijun Deng, Stephane Lafortune +1 more

Orbital Stability of Smooth Traveling Solitary Waves to the Fornberg-Whitham Equation

Certain smooth traveling solitary waves resist small perturbations and keep their form in this shallow-water equation

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The Fornberg-Whitham (FW) equation was introduced by Fornberg and Whitham [Fornberg and Whitham, Phil. Trans. R. Soc. Lond. A (1978)] as a nonlocal model for unidirectional shallow water waves capable of capturing wave steepening and breaking. Despite its similarities with integrable shallow-water equations, the FW equation is not completely integrable. Nevertheless, the FW equation is part of the family of peakon-type models as it supports peaked traveling wave solutions. In this paper, we consider smooth solitary wave solutions to the FW equation. We use a variational approach to show that some are orbitally stable.

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nlin.SI 2026-05-20 Recognition

Weierstrass functions give exact solutions for all coherent couplers

by Graham Hesketh

Complete Weierstrass elliptic function solutions for coherent couplers and their relation to degenerate four-wave mixing

Full complex envelopes for both modes are obtained from arbitrary propagation constants and initial conditions.

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Complete analytic solutions for the coherent coupler with arbitrary propagation constants and self- and cross-phase modulation coefficients are presented in terms of Weierstrass elliptic $\wp$, $\zeta$, and $\sigma$ functions, giving the full complex envelopes for both modes under generic initial conditions. Jensen's coupler emerges as a special case of the general system. The mode solutions contain factors of the form $\exp(r\log R(z))$, where $R(z)$ is a ratio of Weierstrass $\sigma$ functions, giving a multi-valued branch structure that is removable by a gauge transformation. A projection from the three-mode degenerate four-wave mixing system onto the two-mode coupler is identified, and the corresponding degenerate-system solutions are single-valued meromorphic Kronecker theta functions. This connection establishes the coherent coupler as a reduction of a broader class of integrable parametric processes and opens a pathway to leveraging known expansions of Kronecker theta functions for further analysis of nonlinear coupler dynamics.

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cond-mat.quant-gas 2026-05-19 1 theorem

Interactions alone produce Bloch oscillations in continuum quantum gases

by Stefano Scopa, Philip Zechmann +3 more

Generalized Hydrodynamics of Bloch Oscillations in the Absence of a Lattice

Generalized hydrodynamics shows finite-density impurities oscillate with renormalized periods and superimposed center-of-mass drift.

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Objects subjected to a constant force generally increase their velocity over time. This expectation fails whenever their energy is a smooth and periodic function of momentum, resulting in periodic Bloch oscillations instead. Periodic dispersions, typical of lattice systems, can also emerge in continuum media through strong interactions. Here, we study the phenomenon of such Bloch oscillations in the absence of a lattice in a paradigmatic model of integrable quantum gases: the two-component Yang-Gaudin model. We derive a generalized-hydrodynamic theory of Bloch oscillations for a finite density of impurities embedded in a homogeneous interacting background, which we show to persist superimposed to a drift due to the acceleration of the center of mass. Moreover, we show the single-impurity oscillation period is renormalized at finite impurity density when two-magnon bound states are populated. Our results are relevant for ultracold atom experiments, where impurities can be created at controllable densities.

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hep-th 2026-05-19 2 theorems

Classical Wronskians solve zero-momentum sine-Gordon quantum model

by Davide Fioravanti, Marco Rossi

From classical Lax ODEs to quantum integrable theories: the moduli

Moduli-dependent functional relations from sinh-Gordon solutions generate TBA equations and prove Zamolodchikov conjectures

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The general idea of this paper is to start from a classical integrable (partial differential) equation which arises as a compatibility condition for a matrix linear differential problem. For definitiveness' sake, a generalised sinh-Gordon equation depending on $2N-1$ complex coefficients or moduli is considered. Then, the connexion coefficients (Wronskians) of different solutions to this problem satisfy, in the spirit of the Ordinary Differential Equation/Integrable Model correspondence, functional relations, which can be considered, -- if supplemented by suitable asymptotic behaviours --, as identifying a specific state of a quantum integrable model: in fact they are the eigenvalues of extensions of Baxter operators $Q$ and $T$, the transfer matrix. Moreover, Y-system and (implementing the asymptotic conditions) thermodynamic Bethe Ansatz equations originate from them, without any passage through the scattering theory, and providing an invariant parametrisation of the monodromy space. The crucial novelty is the modification of all the relations because of their dependence on the moduli. For zero momentum, they fully describe physically the quantum homogeneous sine-Gordon model, {\it i.e.} scattering amplitudes of gauge fields in $\mathcal{N} = 4$ SYM at strong coupling or their dual null polygonal light-like Wilson loops in $AdS_3$. As a direct consequence of the correspondence, two Zamolodchikov's conjectures, based on previous results, are also proven.

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math-ph 2026-05-19 2 theorems

Dark soliton gas reduces genus step by step in long-time evolution

by Marco Bertola, Deng-Shan Wang +2 more

Arbitrary-genus dark soliton gases in the defocusing nonlinear Schr\"{o}dinger hydrodynamics

The potential matches a genus-N finite-gap solution far to the left and the constant background far to the right, with dynamics showing a 1D

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The defocusing nonlinear Schr\"{o}dinger hydrodynamics supports exact dark solitons under finite density boundary conditions. However, the dark soliton gas, an interacting ensemble of dark solitons, has not yet been studied. In this work, we introduce an arbitrary-genus potential of dark soliton gases by considering the limit of the $\mathcal{N}$-dark soliton as $\mathcal{N}\to \infty$. The large-space asymptotics and long-time evolution of this dark soliton gas potential are analytically investigated through Deift-Zhou nonlinear steepest descent approach. The genus-$N$ dark soliton gas potential approaches the genus-$N$ finite-gap solution as $x \to -\infty$ and the background $1$ as $x \to +\infty$. In the long-time evolution, as the self-similar variable $\xi=x/t$ increases, the gas configuration exhibits a cascade of behaviours, passing from unmodulated and modulated genus-$N$ regions and progressively reducing the genus down to the planar region (unmodulated genus-$0$ region). Notably, the evolution of lower-genus soliton gases can be embedded within that of higher-genus gases, exhibiting identical dynamics within specific regimes. This phenomenon is encoded by the underlying spectra. We also include numerical validations, in perfect agreement with the theoretical predictions.

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hep-th 2026-05-19 2 theorems

Recursion shows Yangian symmetry in deformed sigma models

by Daniele Bielli, Christian Ferko +2 more

The classical Yangian symmetry of Auxiliary Field Sigma Models

A generalized BIZZ procedure produces non-local charges that form a Yangian algebra and preserve Maillet brackets in auxiliary field models.

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Integrable field theories exhibit infinitely many symmetries which underlie their solvability, but the structure of these symmetries can become obscured after performing an integrable deformation such as $\TT$ or an auxiliary field deformation. In this paper, we present a systematic organizing principle for understanding deformed charges and their Yangian structure in a broad class of integrable sigma models and their auxiliary field deformations. We generalize the recursive procedure of Brezin, Itzykson, Zinn-Justin, and Zuber (BIZZ) for generating non-local charges, and give sufficient conditions under which the resulting charges obey a Yangian algebra. We apply these results to many examples of integrable sigma models and their auxiliary field deformations, finding a Yangian algebra and Maillet bracket structure in all cases. This offers a unified explanation for the persistence of Hamiltonian integrability and Yangian symmetry across a wide landscape of deformed sigma models.

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nlin.SI 2026-05-19 Recognition

Darboux transformations construct tau functions for constrained CKP

by Danqi Chen, Jipeng Cheng +1 more

The tau functions of the constrained CKP hierarchy

The method works for both odd and even powers and expresses solutions directly through the auxiliary functions that define the constraint.

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The CKP hierarchy is one important sub-hierarchy of the KP hierarchy, which is quite special due to its tau function. Here we construct the tau functions for the constrained CKP hierarchy $(L^k)_{<0}=\sum_{i=1}^{m}\big(q_{1,i}\partial^{-1}q_{2,i}-(-1)^kq_{2,i}\partial^{-1}q_{1,i}\big)$ with $k$ being odd or even positive integer by using the CKP Darboux transformations.

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nlin.SI 2026-05-18 Recognition

Cross-diffusion system admits time translation and parabolic scaling

by Manjit Singh, Radhika

Lie symmetry classification of a coupled nonlinear cross-diffusion system in radial geometry

Strong nonlinear coupling restricts point symmetries to kernel ones except when constitutive functions satisfy special invariance conditions

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In this work, Lie symmetry analysis is performed on a coupled nonlinear cross-diffusion system with varying cross-section geometry. The system describes two interacting quantities whose material properties, namely the capacity functions and the diffusion coefficients, depend nonlinearly on the dependent variables. The classical Lie invariance criterion produces a set of sixteen determining equations for infinitesimal symmetry generators. The determining equations are solved by first establishing the universal geometric structure of the admitted generators and then classifying the constitutive functions according to their invariance properties in the state space. It is shown that the system always admits time translation and parabolic scaling as kernel symmetries, with an additional spatial translation admitted only in the Cartesian case. Further symmetries, such as translations, scalings, and rotations in the dependent-variable plane, are obtained by making precise structural assumptions about the constitutive functions. The analysis shows that the strong nonlinear coupling in the governing equations prohibits any new point symmetries from arising in the general case, and that larger symmetry algebras are only attainable in degenerate or linearizable special cases. The symmetries obtained in this work are geometrically consistent with parabolic and radial structure of governing equations.

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nlin.SI 2026-05-18 2 theorems

Transcendental curves locate non-Liouvillian integrals for oscillators

by Jaume Giné, Dmitry Sinelshchikov

Integrability of oscillators and transcendental invariant curves

A cofactor-based search reduces the problem to linear equations and proves integrability for two Painlevé-Gambier systems and one oscillator

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In this work we study the integrability of a family of nonlinear oscillators. Dynamical systems from this family appear in different applications from mechanics to chemistry. We propose an approach for finding first integrals and integrating factors, which is based on the construction and classification of transcendental invariant curves whose cofactors are polynomial or rational in one of the variables. We demonstrate that this approach can be efficiently used for finding non-Liouvillian and non-Puiseux integrable dynamical systems. Its application involves finding solutions only of linear algebraic and linear ordinary differential equations. This allows one to study singularities, including essential ones, of the invariant curves in the complex plane. We illustrate this approach by proving non-Liouvillian integrability of two dynamical systems from the Painlev\'e--Gambier classification and non-Puiseux integrability of an oscillator from the considered family. Furthermore, we construct equivalence classes of the first two dynamical systems with respect to nonlocal transformations. We show that among these equivalence classes there are interesting examples of integrable dynamical systems.

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