It has been known for several decades that the third moment of the multifractal spectrum of the harmonic measure for diffusion-limited aggregates is linked to the underlying fractal dimension of the cluster. We demonstrate, using an argument based on the Hastings-Levitov formulation of diffusion-limited aggregation (DLA) in two dimensions, an even stronger link, connecting the universal amplitude of the third moment to the cluster fractal dimension. This argument can be used for both the standard circular DLA as well as DLA in a cylinder (i.e., with periodic boundary conditions).
Quantum droplets (QDs) are self-bound states of ultradilute quantum fluids stabilized by the interplay between the Lee Huang-Yang (LHY) quantum-fluctuation correction and the mean-field interaction, providing a useful platform for exploring macroscopic quantum phenomena. Recent studies on three-dimensional QDs have introduced the concept of bulk modulus and revealed its connection with the breathing-mode frequency, thereby linking the elastic response of QDs to their collective dynamics. Motivated by this progress, we investigate the elastic modulus of one-dimensional QDs. Based on a super Gaussian variational ansatz, we systematically derive the elastic modulus B and analyze its dependence on the interaction strength and particle number. The analytical predictions are further validated by numerical simulations based on imaginary time evolution and the spatial scaling method. We also establish a quantitative relation between the elastic modulus and the eigenfrequency of the breathing mode. In addition, by incorporating corrections to the droplet width beyond the Thomas Fermi approximation, we obtain the dependence of the ratio {\eta} = B/2 on the control parameters g and N. Unlike the three-dimensional case, where the corresponding ratio follows a simple power-law scaling, the one-dimensional system is affected by the soliton-to-droplet crossover, leading to a more intricate dependence of {\eta} on g and N. Our results show that, in the high-particle-number regime, the elastic modulus asymptotically approaches a limiting value determined mainly by the interaction strength, whereas in the low-particle-number regime it depends on both the particle number and the interaction strength.
We investigate the interaction of small-amplitude waves called phonons, with an initially static kink in an exceptional discretization of the $\phi^4$ model that is free of the Peierls-Nabarro potential. Phonons are generated by a localized harmonic source and scattered from one side of the kink. By computing the transmission and reflection coefficients over the entire phonon band, we demonstrate that the scattering properties depend strongly on the lattice spacing. In the weak-discreteness regime ($h<1$), the kink is nearly transparent and phonons are transmitted through it over most of the phonon spectrum. In contrast, for strong discreteness ($h>1$), significant reflection emerges even though the corresponding continuum $\phi^4$ kink is reflectionless. We further show that depending on the frequency of the incoming phonons, the kink experiences negative radiation pressure and is accelerated toward the incoming phonons for all lattice spacings considered, and this effect is much stronger for the strong discretness. The frequency dependence of the kink velocity and energy transfer is explained in terms of resonances associated with Doppler-shifted phonon frequencies and extrema of the phonon group velocity. Our results reveal that strong lattice discreteness can qualitatively modify phonon-kink interactions even in systems where the static Peierls-Nabarro potential is absent.
Dissipative solitons constitute a robust class of self-localized nonlinear states sustained by the dynamic balance between nonlinearity and gain-loss, possessing an intrinsic stability that stems from their fundamental attractor nature. When combined with lattice truncation, this balance gives rise to dissipative surface solitons (DSSs), whose existence and stability are jointly dictated by boundary-induced confinement and non-Hermitian dynamics. In two-dimensional truncated lattices with linear gain and loss, surface localization emerges within gap regimes, where families of DSSs bifurcate from linear surface localized gain modes as the nonlinearity increases. Increasing the number of waveguide rows at the interface enriches the diversity of supported surface modes in both linear and nonlinear regimes. Although multiple DSS families with distinct phase configurations may coexist within the same gap, their dynamical stability is strongly phase selective. These insights establish linear gain-loss engineering as a powerful mechanism for controlling nonlinear surface localization and provide practical guidelines for realizing robust nonlinear surface states in gain-loss-tailored photonic platforms.
Extreme events in financial systems, often captured by indicators such as volatility, remain difficult to identify close to their onset. Volatility shares many statistical properties with other natural, complex systems which experience extreme events, which we explore in this manuscript. We extend the analogy between rogue waves in optical and hydrodynamical systems to financial volatility by identifying rogue-wave-like peaks with similar statistical properties. We use a Schr\"odinger equation where the potential follows the shape of a Kerr nonlinearity to examine the properties of financial volatility indices within a moving time window. We see evidence of Anderson localisation as a rogue peak approaches in the VIX, and show that the numerical gradient of the system's minimum eigenvalue reliably spikes at the onset of an extreme event. We adapt our methodology to simulate the real-time arrival of data, and show that all but one of the VIX's major peaks can be detected given a reasonable amount of history. We then perform two out-of-sample tests, one for the VXO index, and one for the VSTOXX index, and successfully replicate our initial results, identifying all but one major peak (87.5% or 7/8) in both cases. This method of analysis shows considerable promise as a tool for identifying potential financial crises, aiding in their mitigation.
New numerical methods are proposed for the mixing entropy maximization problem in the context of Miller-Robert-Sommeria's (MRS) statistical mechanics theory of two-dimensional turbulence, particularly in the case of spherical geometry. Two of the methods are for the canonical problem; the other is for the microcanonical problem. The methods are based on the original MRS theory and thus take into account all Casimir invariants. Compared to the methods proposed in previous studies, our new methods make it easier to detect multiple statistical equilibria and to search for solutions with broken zonal symmetry. The methods are applied to a zonally symmetric initial vorticity distribution which is barotropically unstable. Two statistical equilibria are obtained, one of which has a wave-like structure with zonal wavenumber 1, and the other has a wave-like structure with zonal wavenumber 2. While the former is the maximum point of the mixing entropy, the wavenumber 2 structure of the latter is nearly the same as the structure that appears in the end state of the time integration of the vorticity equation. The new methods allow for efficient computation of statistical equilibria for initial vorticity distributions consisting of many levels of vorticity patches without losing information about all the conserved quantities. This means that the statistical equilibria can be obtained from an arbitrary initial vorticity distribution, which allows for the application of statistical mechanics to interpret a wide variety of flow patterns appearing in geophysical fluids.
We investigate the Thouless pumping dynamics of nonlinear gap solitons and attempt to realize topological Chern number switching by modulating nonlinear parameters and varying the ramping rate of the relative phase between periodic potentials. We find that gap solitons can undergo nonlinear instabilities accompanied by finite Bogoliubov excitations under near-adiabatic ramping. Such finite Bogoliubov excitations induce the particle loss of the solitons, leading to reversed propagation directions that signals the occurrence of Chern number switching with analyzing the correspondence between soliton chemical potential and Bloch topological energy band. Our findings offer a feasible strategy for manipulating the Thouless pump dynamics of gap solitons mediated by finite Bogoliubov excitations, with implications for topological quantum transport and quantum computing applications.
Predictions for KPII line-soliton interactions agree closely with exact large-N numerical solutions in two test problems.
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We formulate a general system of kinetic equations for a non-stationary two-dimensional gas of elastically interacting line solitons and apply it to the description of a soliton gas governed by the Kadomtsev-Petviashvili II (KPII) equation. We then verify the predictions of the kinetic theory in two analytically tractable problems: the oblique interaction of a KPII line soliton with a one-dimensional soliton condensate of the Korteweg-de Vries equation, and the interaction of a trial KPII soliton with a monochromatic KPII soliton gas. In both cases, we compare the analytical results with direct numerical simulations obtained by constructing two-dimensional soliton gases via exact KPII $N$-soliton solutions for large $N$, using appropriately chosen random distributions of soliton parameters. The comparison demonstrates excellent agreement, thereby providing strong validation of the proposed kinetic theory of 2D non-equilibrium soliton gases.
Amplification squeezes biphotons while Kerr nonlinearity adds an anharmonic potential, producing collapse-revival dynamics.
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Nonlinear photonic time crystals (PTCs) can support solitons inside momentum k gaps, where the amplification of k gap modes is saturated by Kerr nonlinearity, forming spatially homogeneous but temporally localized excitations. Yet their quantum nature remains unclear. Here we quantize nonlinear k gap dynamics of PTCs and show that k gap solitons are represented by biphoton Fock ladder states. K gap amplification drives two-mode squeezing of the biphoton, while Kerr nonlinearity generates an anharmonic potential along the biphoton Fock ladder that balances this squeezing process, creating a finite biphoton number turning point and giving rise to quantum collapse and revival dynamics and nonclassical phase space interference. We further analyze how photon loss and dephasing reshape the biphoton statistics of quantized k gap solitons. Our results establish a biphoton Fock space description of k gap soliton quantization and provide a framework for studying quantum nonlinear excitations and entangled light generation in photonic time crystals.
In the present work we analyze traveling and dispersive shock waves of a two-dimensional Fermi-Pasta-Ulam-Tsingou lattice. In the first part of the paper, using variational techniques we prove the existence of both periodic and solitary traveling waves for convex potentials. In the case of unimodal profiles we are able to remove the assumption of convexity. The variational formulation also provides a natural algorithm for the numerical computation of traveling waves, which we use to explore both solitary and periodic traveling waves. The numerical computations are compared with analytical approximations based on the derivation of the KdV equation for quasi-one-dimensional propagation. In the second part of the paper, we focus on dispersive shock waves (DSWs), which are expanding modulated waves that connect states of different amplitude. In particular, we focus on line DSWs, which are constant along one direction and propagate in the direction orthogonal to which it is constant. Such solutions form when subject to quasi-one-dimensional jump initial data. We find that while the shape of the DSW depends on the direction of travel, properties such as the speed and amplitude do not. The systematic numerical study of the line~DSWs is then compared to those predicted by the KdV equation along the line of propagation. Key characteristics of the DSWs, such as the speeds of the trailing and leading edges, are investigated for various jump heights, yielding good agreement between simulation and KdV approximation in the limit of vanishing jump height. Finally, we apply the DSW fitting method to study the trailing and leading edge characteristics of the DSW, finding even better agreement to the numerics when compared to the KdV prediction. The KdV prediction and DSW fitting predictions agree in the limit of small jump height.
Self-injection locking (SIL) of a diode laser to a high quality-factor microresonator has recently become increasingly important in hybrid integrated photonics, providing access to compact sub-Hz linewidth lasers. It was also shown to facilitate the access to dissipative Kerr solitons - the key to a low-noise coherent frequency comb on a photonic chip. However, the existence and stability ranges of SIL soliton states in experimentally controlled parameters are still not fully understood. Here we study the bifurcation structure of solutions in a model of soliton SIL in the weak-backscattering limit. We show that SIL produces soliton-number-dependent existence ranges of multi-soliton solutions in free-laser detuning and feedback phase parameters. We identify exclusive single-soliton existence regions and demonstrate dynamical access to single solitons in this region by direct numerical simulations using prescribed parameter sweeps.
Simulations of Aβ-Tau interactions locate multiple steady states, offering one account for resilient cases despite heavy pathology.
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Alzheimer's disease (AD) is characterized by the accumulation of Amyloid-$\beta$ ($A\beta$) plaques and hyperphosphorylated Tau proteins. However, many individuals exhibit substantial $A\beta$ and Tau pathology without developing dementia, suggesting that disease progression may depend not only on pathological burden but also on the spatial organization of these proteins. Motivated by this observation, we adapt Gray-Scott reaction-diffusion model to investigate pattern formation arising from the interactions between $A\beta$ and Tau. %
To systematically identify stable spatial configurations, we employ a Companion-Based Multi-Level Finite Element Method (CBMFEM) on both two-dimensional domains and anatomically realistic cortical surface meshes. Numerical simulations reveal a rich landscape of multiple steady-state solutions, which are subsequently classified into representative pattern phenotypes using principal component analysis and clustering techniques. The results demonstrate that the coupled $A\beta$--Tau system admits numerous stable spatial patterns rather than a single pathological endpoint. %
These findings provide a potential mathematical framework for understanding the heterogeneity of Alzheimer's disease and the existence of cognitively resilient individuals despite significant pathological burden. More broadly, the proposed framework suggests a pattern-based therapeutic paradigm in which disease dynamics are guided toward favorable stable states rather than solely targeting the elimination of pathological proteins.
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.
Based on the original Miller-Robert-Sommeria theory, we explicitly compute a statistical equilibrium of two-dimensional turbulent flow on a sphere for a generic initial vorticity field introduced in a previous study. The macroscopic vorticity field corresponding to the obtained statistical equilibrium has a quadrupole structure. The resulting quadrupole structure is topologically consistent with the final state of the long-term time integration of the vorticity equation. However, the statistical equilibrium does not predict the formation of concentrated vortices as seen in the time integration. We also calculate statistical equilibria for the initial vorticity field with a planetary vorticity term, and find a change of statistical equilibria from quadrupole states to zonally symmetric states as the angular velocity of the sphere increases. The quadrupole statistical equilibria show nearly linear relations between the macroscopic vorticity and the macroscopic stream function, implying that higher-order Casimir invariants are virtually ineffective even when all Casimir invariants are considered. The discrepancy between the equilibria and the time integration results emphasizes the importance of mixing barriers, which prevent the relaxation of the evolving vorticity field to the statistical equilibria and allow the point-vortex-like dynamics of coherent vortices to persist.
When implementing the method of multiple scales, one is traditionally instructed to treat the slow and fast time scales as if they were independent. Despite the intuitive motivation and the effectiveness of this perturbation method, one cannot failt to notice that these two scales relate to the same unique variable, so independence can only be formal. How sensible is it, then, to split a variable asymptotically into two (or more) independent ones? In this paper, we elucidate this issue with Van der Pol's equation, one of the simplest weakly nonlinear oscillators, as well as a simple example of a Hopf bifurcation. The discussion involves carrying the multiple-scale analysis up to arbitrarily large order and dealing with the divergent character of the resulting asymptotic series. Using the technique of optimal truncation, we re-connect the two scales. Specifically, we show that an initial translation of the fast coordinate leads to a non-trivial, exponentially small, phase shift that depends on the slow coordinate. This phase shift breaks the independence of the slow and fast scales and is found to result from the nonlinearity. Numerical simulations confirm its existence, as well as the predicted scaling. The calculation is carried out in sufficient detail to provide confidence in the generality of our result, both in its essence and in its form. In particular, we find strong indications that a Hopf bifurcation with a quadratic nonlinearity would lead to the same phenomenon, but with a larger magnitude.
The process of excitation of degenerate optical parametric oscillations via bichromatic pump is studied numerically in normal-dispersion photonic-crystal microresonator. It is demonstrated that the photonic-crystal structure with two split modes placed symmetrically at the particular interval from the pumped modes provides significant reduction in pump power threshold for the considered process. The parameter range for this phenomenon is determined. Introduction of mode splitting at the signal mode located in the center between pumped modes leads to an increase in the generation threshold.
Straight dark solitons in two-dimensional (2D) quantum fluids usually decay by transverse modulational instability, with no intrinsic suppression in contact-interacting Bose--Einstein condensates (BECs). We theoretically show that anisotropic long-range interactions in a quasi-2D dipolar BEC stabilize an embedded straight soliton, with spontaneous stripe order providing stronger pinning. The excitation spectra show that the lowest transverse solitonic branch remains gapped, while stripe-supersolid density modulation further hardens this branch and increases the soliton bending stiffness, penalizing transverse deformation. Accessible in current $^{166}$Er and $^{164}$Dy platforms, these results establish interaction-driven protection for straight dark solitons in structured quantum fluids.
The quantum geometry of electronic, photonic, and atomic lattice systems quantifies the distance in Hilbert space between Bloch states at neighboring lattice momenta. This quantity has profound implications for flat-band systems especially, characterizing surprising behavior such as superfluidity and superconductivity when the group velocity is zero and no transport would be expected for non-interacting particles. However, when the band is not flat, the effects of quantum geometry are often intertwined with and partly masked by the band dispersion. Here, we show that in weakly interacting bosonic systems in the critical dimension (i.e., two dimensions for Kerr nonlinearity), the deviation from critical behavior due to the presence of the lattice is governed by the quantum geometry, which is directly proportional to the fourth-order dispersion. Furthermore, we identify the family of continuous lattice potentials that saturates the bound on the quantum metric for a given effective mass tensor.
We introduce the Quantum Split-Step Fourier (QSSF) algorithm for nonlinear optical waveguides, a numerical framework that combines split-step propagation of the nonlinear Schr\"odinger equation with a commutator-preserving Bogoliubov evolution of Gaussian quantum fluctuations. The method propagates the classical mean field together with the Bogoliubov matrices $U$ and $V$, from which reduced second moments, covariance matrices, symplectic eigenvalues, and entropic measures are constructed for arbitrary spectral windows. Applied to soliton-driven resonant radiation, QSSF shows that the selected radiation band acquires a steadily increasing von Neumann entropy and a corresponding loss of purity, quantifying its entanglement with the rest of the spectrum in the lossless Gaussian setting. The analysis also reveals a surprisingly pronounced low-dimensional structure: although the radiation occupies many Fourier bins, its reduced Gaussian state is dominated by only a few Williamson modes. QSSF therefore provides a practical information-theoretic diagnostic for quantum correlations in nonlinear frequency conversion, supercontinuum generation, and multimode squeezed-light formation in ultrafast waveguide platforms.
Demographic noise generates stochastic Turing patterns even when reaction-diffusion systems are deterministically stable. We show analytically and verify numerically in the Levin-Segel model that temporal integration of configurations reveals emergent large-scale organization. The intensive number variance in a window of size $R \gg 1$ approaches a finite reaction-kinetic floor as $1/R$, over a spatial range growing by orders of magnitude near the Turing instability. This yields an effectively hyperuniform, fine-tuning-free regime previously unidentified in non-conserved multispecies stochastic systems.
Frequency combs are discrete, equally spaced, phase-coherent spectral lines that emerge from nonlinear mode coupling in physical systems. We show that the incommensurate fractional-order financial model of Huang, Li, Ma, and Chen, whose Caputo derivatives encode macroeconomic long-range memory, generates an analogous structure in its steady-state spectrum. The comb appears only over specific values and ranges of the saving amount $a$, the investment cost $b$, and the demand elasticity $c$, outside which the spectral lines lose their equal spacing. It persists across extended parameter regimes and stays invariant to perturbations in the initial interest rate $x_0$ and investment demand $y_0$, while distinct spectral regimes appear at different initial price levels $z_0$. The comb is generated only when the fractional-order exponents $q_1$, $q_2$, and $q_3$ associated with interest rate, investment demand, and price index are above the critical threshold values. At even higher values of these exponents, the frequency comb transitions into chaos. These findings show that the long-run cyclic structure of a memory-bearing financial economy organises into a discrete, deterministic spectral fingerprint rather than a stochastic continuum.
We show that an oscillon can be understood as a localized discrete resonant (non-normalizable) mode. Specifically, oscillon in the vacuum arises from the threshold mode, which because of nonlinearity gets localized. Following this idea, we find {\it wobblerons} - nonlinear excitations of kinks, that is, oscillons-kink bound state. Now, the oscillon can also originate in an antibound mode, i.e., a discrete, positive energy but non-normalizable mode.
Coupling temperature response to annular Laguerre-Gauss ansatz captures mode-order and nonlocality dependence without solving the full spati
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Nonlocal optical nonlinearities arising from the thermorefractive effect provide a long-range material response determined by heat diffusion and absorption. In graded-index media, this nonlocality fundamentally alters modal interactions, yet its accurate modeling remains computationally demanding when starting from the full spatial nonlinear Schr\"odinger equation. In this work, inspired by the nonpolynomial Schr\"odinger equation (NPSE) framework, we extend the dimensional reduction techniques to incorporate thermally mediated nonlocal nonlinearities. By coupling the optical field to an equation for the temperature-induced refractive index change, and employing a variational ansatz based on Laguerre--Gauss modes of the annular kind, of arbitrary azimuthal order, we derive explicit analytic expressions for the variational equations. The resulting effective model captures the dependence of the nonlinear interaction on mode order and degree of nonlocality, providing a tractable reduced description of the dynamics in thermal nonlocal media.
Recent advances in the work with ultracold condensates of polar molecules have enabled the realization of highly tunable self-bound quantum droplets (QDs), with the help of dual microwave fields dressig the dipole-dipole interactions (DDIs) It has been reported that symmetry properties and the equilibrium phase diagram of such QDs can be controlled by parameters of the two microwave fields. However, the effect of these fields on the formation and dynamics of the QD has not yet been systematically explored. Here we address self-bound QDs in a regime dominated by non-axisymmetric DDIs and governed by the extended Gross-Pitaevskii equation with the Lee-Huang-Yang corrections. Within this framework, we identify the existence region of the self-bound QDs and characterize their chemical potential, total energy, effective volume, peak density, and geometric anisotropy. The results reveal a pronounced nonmonotonous dependence on the non-axisymmetric DDI strength, whereas the increase of the number of particles in the condensate leads to tighter bound and more anisotropic QDs. Furthermore, reducing the s-wave scattering length drives a transition from stable self-bound states to the collapse. Collisions between QDs moving along different directions reveal a strong directional dependence, with outcomes ranging from quasi-elastic rebound and merger to fragmentation.
This work is devoted to data-driven identification of discrete breathers in numerical simulations of a two-dimensional crystal lattice using locally sampled wave data. Different lattice wave datasets are considered, with data collected from regions of different shapes and sizes defined by the lattice particles in mechanical equilibrium. Specifically, in addition to regions with a regular hexagonal shape, one- and quasi-one-dimensional regions reflecting the quasi-one-dimensionality of discrete breathers in two-dimensional hexagonal crystal lattices are proposed. To improve numerical efficiency, dataset dimensionality is reduced using Principal Component Analysis, and highly accurate Support Vector Machine classifiers are trained to distinguish between linear and nonlinear wave data. The obtained classifiers, together with the sliding window method, are applied to detect localization regions in two-dimensional hexagonal crystal lattice numerical simulations. High-precision algorithms for detected localization region segmentation and localized wave directionality estimation within the detected regions are further proposed, and their performance is evaluated. The presented methods are successfully applied to detect localized waves and their collision regions, as well as their directionality, performing a numerical study of stationary and traveling two-dimensional discrete breather interactions. Qualitatively better results are obtained when considering wave-data collection regions respecting the quasi-one-dimensional nature of two-dimensional discrete breathers in the hexagonal crystal lattice model.
We are concerned with the geometric structures of pseudo-sonic curves in two-dimensional self-similar solutions for the Euler equations for potential flow, allowing for non-uniform supersonic states. Mathematically, the governing second-order potential flow equation is of mixed hyperbolic-elliptic type, with degeneracy occurring along the pseudo-sonic curve. In this paper, we develop rigorous analytical approaches to analyze the geometric structures of pseudo-sonic curves in such self-similar solutions. We first show that the pseudo-sonic curve is necessarily a circle if the pseudo-velocity at each point is a normal to the curve. We then analyze the general case in which the pseudo-velocity on the pseudo-sonic point is not a normal to the curve, and study the geometric properties of streamlines in a neighborhood of the pseudo-sonic curve. Next, we establish two theorems that provide sufficient conditions ensuring that the pseudo-velocity at a pseudo-sonic point is normal to the curve, under natural assumptions on the local behavior of the solution. These results yield a precise characterization of the geometry of pseudo-sonic curves. Finally, we apply the developed theory to the shock reflection-diffraction problem with non-uniform incoming flow. We prove that the pseudo-sonic curve must be an arc if the solution is a $C^2$-small perturbation, either in the pseudo-supersonic or pseudo-subsonic region, of a solution with uniform incoming flow. In particular, the density and velocity must be constant, corresponding to the radius and the center of the pseudo-sonic arc, respectively. Moreover, we prove that the solution is $C^{2,\alpha}$-regular in the pseudo-subsonic region up to the sonic arc (except at point $P_1$). The techniques and ideas developed in this paper are expected to be applicable to other nonlinear problems involving similar mixed-type degeneracies.
Small-size, thin-wall, and large-multiplier regimes each yield approximate solutions for false-vacuum decay in a toy scalar model.
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We revisit the topic of false vacuum decay in field theory. We focus on a toy model of a real massive scalar field with an unstable quartic potential. This model has a false vacuum, and decay out of the false vacuum can be described via the method of constrained instantons, which work by introducing a constraint on the path integral. We identify and develop three different asymptotic limits which enable analytic construction of approximate {constrained} solutions. The first, in which the constrained solution is small compared to the inverse mass of the scalar field, is an application of the perturbative methods of Affleck, although we re-derive the main results and identify several terms which were previously neglected. Second, for very large constrained solutions we adapt the thin-wall approximation of Coleman. However, we find that the large instanton limit does not always exist. In this case we identify another useful limit, in which the Lagrange multiplier used to implement the constraint is large. In this limit, the solution's scaling with the parameters may be found via dimensional analysis and an exact solution is obtained with a single numerical computation.
In the focusing Manakov system, multiple modulation instability (MI) branches coexist on the same plane wave background, so the usual weak periodic modulation cannot selectively excite a single vector Akhmediev breather (AB). Here we propose an eigenvector-based initial perturbation scheme that constructs the initial condition as a plane wave plus Fourier modes whose coefficients follow the perturbation eigenvector of a selected MI branch, enabling controllable high-fidelity excitation of desired vector ABs. Numerical simulations show near-100\% fidelity with the exact AB solution. The underlying mechanism is eigenvector-controlled mode selection. The initial seeding of the target MI branch through the chosen eigenvector, together with the non-Hermitian coupling inherent in the linearized MI dynamics, ensures that the targeted unstable mode dominates the early linear stage and thereby dictates the breather type. This eigenvector-based control succeeds in gain-balanced regimes and when the targeted branch has a sufficient gain advantage. The proposed method provides a simple and robust framework for controllable generation of vector ABs over a broad parameter range, highlighting the key role of eigenvector selectivity in multi-component nonlinear systems.
We show that compartmentalizing a single molecular species into intracellular and extracellular fields, and coupling them through membrane transport or nonlinear basal production rates, can produce diffusion-driven (Turing) instabilities. By linearizing the two-field system, we derive the corresponding Turing conditions under which such instabilities may arise. We present three biologically motivated examples that satisfy these conditions and demonstrate the resulting spatial patterns through numerical simulations. These results indicate that tissue compartmentalization alone might enable pattern formation traditionally attributed to multi-species systems.
Kerr frequency combs have recently emerged as an exciting new photonic technology, with applications across science and engineering. Their formation within driven optical resonators that possess a Kerr nonlinearity is enabled through the rich landscape of localized nonlinear dissipative structures intrinsic to these systems. This article offers a comprehensive review of the physics that underpins these nonlinear comb-generating structures. Particular attention is placed on bright temporal cavity solitons and nonlinear switching waves -- the canonical stable comb-generating states in the anomalous and normal dispersion regimes, respectively. Written as both a review and tutorial, the article also includes an in-depth treatment of the numerical methods required to simulate driven Kerr resonators, alongside a comprehensive discussion of the laboratory techniques used to experimentally realize and characterize Kerr combs.
Supratransmission is examined in nonlinear lattices with purely nonlinear coupling, extending the phenomenon to systems that lack a linear pass band. In contrast to standard lattices with mixed linear-nonlinear interactions, the present model has no linear spectrum, so energy propagation arises entirely from nonlinear effects. Asymptotic analysis yields a discrete $p$-Schr\"odinger (DpS) equation that {provides an accurate description in the weak- and intermediate-coupling regimes and offers qualitative insight in the strong-coupling regime}. Perturbation provides analytical approximations for the critical driving amplitude, explicitly showing its dependence on the driving frequency, coupling strength, and the nonlinearity exponent $p$. The analysis identifies a non-trivial dependence of the critical amplitude on $p$, with distinct trends in different coupling regimes. Numerical continuation and direct simulations {validate the theory in regimes where the asymptotic reduction is applicable and show good agreement across a wide range of parameters}. The results establish supratransmission in fully nonlinear lattices and clarify the associated energy-transport mechanisms, with relevance to mechanical lattices, tunable metamaterials, and nonlinear optical arrays.
In the present work, we present a number of localized wave patterns that are theoretically analyzed and
numerically illustrated to be observable within the
widely applicable paradigm of the FPUT lattice. In particular, we derive a modified KdV equation from the FPUT lattice, which admits a variety of localized waves including these exact rational solutions representing rogue-wave profiles, solitons and breathers on the top of not only homogeneous,
but also periodic elliptic function traveling-wave background. We utilize these exact solutions of the modified KdV reduction to construct consistent initial conditions for the FPUT lattice and perform time stepping of the latter. Relevant comparisons between these numerical solutions of the FPUT lattice and their associated analytical counterparts have been conducted to demonstrate good performance of the derived modified KdV reduction in approximating distinct localized wave structures from the FPUT lattice.
This approach paves the way for importing a number of
quasi-continuum waveforms to the FPUT lattice and the
potential associated physical experiments, including recent
ones in mechanical metamaterials.
Simulations show quantitative shifts in critical velocities and resonance windows even when kink profiles remain similar.
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We investigate kink antikink collisions in higher order scalar field theories described by the higher order models and their shifted periodic extensions. Both classes of models possess three degenerate vacuum states and support topological kink solutions with asymmetric profiles and algebraically decaying tails. By extending conventional polynomial potentials across multiple spatial sectors, we construct shifted periodic high order field theories and examine how this modification affects the scattering dynamics of topological defects. The primary objective of this study is to provide a comparative numerical analysis of kink collisions in the standard and shifted periodic versions of these higher order models. Using direct numerical simulations, we determine the critical velocities that separate capture from escape regimes and identify resonance structures associated with energy exchange between translational and internal vibrational degrees of freedom. Particular attention is devoted to the emergence of escape windows, quasi-fractal patterns, and the role of algebraic tails in shaping the collision outcomes. Our results demonstrate that, although the conventional and shifted periodic models exhibit similar kink antikink configurations, important quantitative differences arise in their critical velocities, resonance structures, and scattering characteristics. The findings further confirm that both classes of models support resonant energy transfer mechanisms analogous to those observed in lower order theories, while simultaneously exhibiting novel features associated with higher-order interactions and long range effects. These results contribute to the growing understanding of nonlinear excitations in scalar field theories and provide new insights into the dynamics of topological solitons in shifted periodic systems
Energy localization appears in secular terms of damped disordered Duffing chains when driven away from primary resonance.
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This paper proposes a framework for energy localization in nonlinear oscillator chains operating in non-fundamental resonance, with emphasis on superharmonic regimes. The system is modeled as a metastructure composed of Duffing oscillators with both linear and nonlinear coupling, incorporating disorder-induced periodicity breaking. Under the assumption of strong excitation, the method of multiple scales is employed to derive the governing equations for soliton dynamics. At first-order perturbation, the classical form of the Nonlinear Schr\"odinger Equation emerges, whereas second-order analysis yields a previously unreported equation arising from the restitution of time scales. Analytical and numerical results demonstrate the nucleation of solitons in both hardening and softening regimes, based on two approaches: direct time-domain simulations from initially motionless states and numerical continuation in the frequency domain. A key finding is the distinct role of phase in superharmonic resonances compared to the primary resonance; specifically, the coexistence of multiple frequency components in the steady-state response precludes interpreting the soliton directly as a displacement envelope. Instead, the resulting secular terms captures the soliton associated with the resonant contribution, while transient components remain present under superharmonic excitation. Furthermore, robustness against disorder uncertainty is assessed by determining the tolerance levels that preserve the phenomenon. These results support the development of vibration control strategies aimed at mitigating the increase in resonant frequencies associated with the geometric downscaling of mechanical systems.
Among the most intriguing properties of honeycomb lattices is the presence of Dirac points that typically emerge in pairs, which can be destroyed by physical effects breaking certain symmetries of the system and leading to nontrivial band topology. We propose a nonlinear microcavity system supporting condensate of exciton-polaritons, where simultaneous breakup of inversion and time-reversal symmetries results in unusual spectrum with unpaired Dirac cones, profoundly affecting the properties of unidirectional edge states. Realized as an array of microcavity pillars, the inversion symmetry is broken by fission of pillar belonging to one of sublattices of honeycomb array into three pillars, while time-reversal symmetry is broken due to interplay of Zeeman splitting in the external magnetic field and spin-orbit coupling. Despite the absence of complete spectral gap, unidirectional edge states may still emerge that can circumvent array corners. Resonant optical pumping leads to reach bistability effects and allow selective excitation of the edge states. We obtain first example of stable localized dissipative edge soliton that circulates along the periphery of insulator over indefinitely long times without radiation. Our results suggest a new platform for nonlinear topological photonics and reveal nontrivial interplay between unpaired Dirac cones and nonlinear effects.
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.
Three-dimensional (3D) optical solitons bear far stronger relevance to multi-dimensional nonlinear dynamics prevalent in complex physical, chemical and biological systems than conventional 1D solitons, and they support far richer and more intricate phenomena arising from their spatiotemporal degrees of freedom. However, real-time recording of 3D soliton evolution with simultaneous spatiotemporal resolution remains a critical challenging. Here, we demonstrate long-term real-time visualization of 3D soliton dynamics with spatiotemporal resolution using high-speed photodetectors combined with joint space- and time-division multiplexing. We visually capture complex transient behaviors of 3D solitons in a multimode fiber laser and, by integrating our method with the time-stretch technique, simultaneously record pulse-resolved beam and spectral evolutions. We observe that during the birth of 3D solitons, the highly multimode beam stabilizes for a substantial interval prior to spectral broadening, indicating that a large number of transverse modes have already locked before longitudinal mode proliferation. These findings highlight the critical importance of real-time spatiotemporal visualization for advancing ultrafast multimode laser design and delivering new insights into high-dimensional nonlinear dynamics.
The article is devoted to the problem of decay of initial stationary state for the Volterra lattice with zero boundary condition. We show that this process is asymptotically self-similar and calculate the propagation velocity of the decay wave, the leading terms of the asymptotics and corrections, in the main and transition sectors of the wave.
Nonzero azimuthal charge breaks symmetry above a critical amplitude, sending out two-dimensional localized excitations.
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We investigate energy supratransmission in a boundary-driven two-dimensional nonlinear Schr\"odinger equation with a central hole. Harmonic forcing with azimuthal modulation generates standing-wave states whose existence and stability depend on the driving amplitude, the inner radius, and the imposed azimuthal charge. Bifurcation analysis shows that small inner radii produce strongly confined states with higher destabilization thresholds, whereas larger radii yield broader profiles and smoother transitions between stable and unstable branches. The cubic--quintic and saturable models exhibit similar qualitative behaviour but differ quantitatively in their critical amplitudes and parameter dependence. A variational approximation captures the dependence of the critical drive on the azimuthal charge and nonlinear parameters, and clarifies how the nonlinear response shapes the stationary states near the turning point. Time-dependent simulations show that supratransmission occurs through the emission of localized pulses, with nonzero azimuthal charge triggering symmetry breaking and producing two-dimensional localized excitations (pearls). Isosurface plots provide a complementary view of the resulting radial and angular excursions. These results establish a quantitative framework for supratransmission in two-dimensional geometries and are relevant to driven nonlinear systems in optics, Bose--Einstein condensates, and structured media.
This work introduces a biophysical formalism to describe the spatiotemporal evolution of the chemical profile in tissues, with the novelty of modeling tissue compartmentalization and the mechanism by which cells maintain the system far from thermodynamic equilibrium via production and/or degradation of substances. The models were derived from conservation laws, chemical kinetic theory, and geometric constraints, while considering fundamental properties of tissues to connect theoretical modeling with experimental observations. In a morphogenetic context, each morphogen is described by two coupled reaction-diffusion equations, representing intra- and extracellular dynamics, linked through membrane transport processes such as nonlinear, cross, and anomalous diffusion. We explore the models' morphogenetic potential through diffusion-driven instabilities and discuss how natural tissue heterogeneities influence Turing instabilities and self-organized phenomena. The mathematical structure reveals that two-morphogen systems can produce Turing patterns with multiple characteristic length scales, while the system's dimensionality enables chaotic behavior in well-mixed dynamics. Moreover, due to domain coupling, Turing instabilities are allowed for single-morphogen systems. We used Schnakenberg kinetics to demonstrate that Turing patterns arise even when the activator diffuses faster than the inhibitor (d$<$1), thereby expanding the parameter space for pattern formation. Our results suggest that tissue spatial structure has important consequences for Turing instability mechanisms, in some cases weakening the usual conditions for its emergence while widening the possible patterns it can produce. The proposed framework offers a minimal mathematical basis to explore emergent dynamics in biological and synthetic contexts, with potential applications in developmental biology and tissue engineering.
Gradient-based inversion of reaction-diffusion systems is typically approached via surrogate models or physics-informed neural networks (PINNs), while the most direct route, backpropagation through the PDE's structure itself, has largely been avoided. We pursue this direct route as a diagnostic probe, backpropagating a steady-state loss through unrolled Gray-Scott simulation to recover its parameters, with no surrogate or neural-network augmentation. Optimization fails to converge, and plotting the landscape directly locates the failure in its geometry -- flat plateaus with no gradient signal, bounded by sharp cliffs that align with bifurcation boundaries -- a structure that recurs across loss functions and is inherited however the gradients are routed to parameters. Reading this minimal setup as an ablation of PINN, we disentangle each component's role: with the neural network fixed, the residual loss is quadratic in the PDE parameters and yields a smooth landscape, so it alone already avoids the pathology, by implicitly encoding the full PDE dynamics across all initial conditions. The neural network, for its part, cannot repair an ill-posed parameter subspace, and so serves only to complete the observed data -- a division of labor not previously made explicit. These findings carry concrete design implications for PINN-type methods and a broader heuristic on when added dimensions actually help.
We study how time-periodic, spatially localized states are born from the linear spectrum of a \emph{finite} lattice as the nonlinearity is switched on. In earlier work we treated this question for a diatomic chain with on-site nonlinearity and developed a framework that continues a near-edge linear mode in amplitude and controls the resulting perturbation series uniformly in the chain length. The present paper shows that the same framework applies to the more difficult case of Fermi--Pasta--Ulam--Tsingou (FPUT) interaction nonlinearity. The key is a structural relation between the FPUT and on-site nonlinearities, which allows the estimates obtained in the on-site setting to be transferred to the FPUT setting. As before, the analysis yields a quantitative radius of convergence, $\eps=\Theta(1/\sqrt{n})$ for a chain of length $2n$, below which the near-edge mode stays extended and above which the orbit localizes and its frequency leaves the band. The diatomic chain is used only as a test case; both the formation mechanism and the method are model-independent and are expected to extend to other short-range nonlinearities and to higher dimensions.
We investigate the different oscillatory modes, namely, complete synchronization, inphase synchronization, antiphase synchronization and desynchronization in a one-dimensional anisotropic Heisenberg ferromagnetic spin chain consisting of a large number of spins. By solving the associated Landau-Lifshitz-Gilbert-Slonczewski equation for the spins we show the simultaneous existence of the above mentioned oscillatory modes in the spins. We observe that when the number of the spins is large the synchronization is lost between the spins; however, we identify that the field-like torque is able to induce synchronous oscillations of the spins in the chain again. We also confirm the agreement of the numerically obtained values of the frequency of the inphase synchronized oscillations with the analytically obtained values.
We present a rigorous theoretical framework for symmetry breaking and quantum irreversibility arising from stochastic Ito field reversal within a cubic-quintic nonlinear Schrodinger equation (CQ-NLSE) formalism. Starting from three physically motivated considerations, forward and backward nonlinear stochastic differential equations are derived via the Ito calculus. Kinematic time-reversal is shown to be fundamentally incompatible with the Ito stochastic structure, yielding the universal asymmetry-coupling parameter of 2/3. An energy-driven collapse operator proportional to the product of noise strength, local probability density, and excitation energy squared is introduced, amplifying the collapse in high-density, high-excitation regions. Exactly bright soliton solutions are obtained for a quasi-one-dimensional BEC of attractive Li-7 atoms, with forward and backward amplitude ratio of 1.870. Heat map analysis of the parameter planes reveals that the forward collapse operator grows monotonically in time while the backward counterpart decays, achieving a ratio approximately 1030, sharply distinguishing this framework from conventional symmetric collapse models.
In this paper, we develop and analyze exponential time differencing (ETD) schemes for a phase-field model of multicomponent membranes proposed in our previous work \cite{luo2025ohta}, in which membrane deformation is governed by a force-balance phase-field equation and protein segregation is described by a membrane-associated Ohta-Kawasaki (OK) dynamics. For a fixed phase-field membrane, we introduce a geometry-adapted operator splitting method based on the localization function, which reformulates the surface OK dynamics into a form suitable for ETD integration. The resulting first- and second-order ETD schemes, combined with finite-difference spatial discretization, are rigorously proved to satisfy a discrete maximum-bound principle and unconditional energy stability. For the coupled system, we construct stabilized ETD schemes in an FFT-based spectral framework, treating stiff linear terms exactly and nonlinear mechanochemical couplings explicitly. A narrow-band implementation further reduces the computational cost by restricting surface calculations to the diffuse membrane region. Numerical experiments confirm the predicted temporal accuracy, maximum-bound preservation, and energy decay for the fixed-membrane OK problem, and demonstrate stable and efficient three-dimensional simulations of protein-driven pattern formation and membrane deformation.
Thouless pumping provides a paradigmatic platform for studying the effects of interactions on topological transport in periodically driven systems. However, most studies have been constrained by adiabatic conditions, which preclude exploration of interaction-driven novel topological states at high driving frequencies. Here, we experimentally investigate the interplay between interaction and modulation frequency in Thouless pumping realized in a periodically modulated lattice in momentum space of atomic Bose-Einstein condensate. We observe fast Thouless pumping of matterwave solitons at intermediate interactions, with no counterpart in the non- or weakly interacting regimes. Beyond the boundary of topological phase transition induced by interaction, nonadiabatic quantized pumping of solitons emerges at high modulation frequencies over a broad interaction range, in good agreement with theoretical calculations, while the solitons remain trapped in the low-frequency adiabatic pumping regime. Our work opens new avenues for accelerating topological transport in driven quantum systems and engineering fast topological devices.
Solitons in integrable systems exhibit a dual wave-particle character, yet their identification as individual objects becomes ambiguous during interactions, where they deform and delocalize. We develop a microscopic, field-based description that resolves this issue by introducing exact space-time fields of charge, current, and velocity derived from the inverse scattering transform (IST) of the Korteweg-de Vries (KdV) equation. This framework enables individual KdV solitons to be tracked throughout interactions, providing a quantitative description of their trajectories and deformation beyond the asymptotic regime. We show that the dynamics of $N$ interacting solitons can be formulated in terms of $N$ independent continuity equations, in which interactions are encoded in initial correlations that are subsequently propagated. From this microscopic viewpoint, effective velocities and hydrodynamic behavior emerge upon coarse graining, recovering kinetic theory of soliton gases and Generalized Hydrodynamics as scaling limits. Our results establish a direct connection between the wave-based IST formalism and particle-like emergent descriptions, offering a unified framework for soliton dynamics across scales.
From spin ice and crumpled paper to cold atoms lattices and metamaterials, geometrical frustration occurs generically whenever local constraints cannot be satisfied all at once. The result is a ground state degeneracy, where many equivalent states, each of which contains unsatisfied constraints, coexist. Here, we introduce dynamical frustration, where the ground state degeneracy makes way to a non-reciprocal self-oscillating state instead. To create dynamical frustration, we construct metamaterials that are driven parametrically in time and modulated in space. The parametric pumping leads to period doubling and in turn to a discrete symmetry-breaking. This symmetry breaking, together with the spatial modulation enforces the existence of topologically protected phase dislocations, which propagate unidirectionally with a spontaneous phase that breaks a continuous symmetry. Tesselating 1d frustrated loops, one obtains a 2d metamaterial where phase dislocations self-organize into globally synchronized non-reciprocal phase defects. We expect dynamical frustration to be broadly applicable at any scale, from cold atoms and superconducting circuits to acoustics and RF circuits -- anywhere where space-time modulation can be pushed beyond linear stability.
Real complex systems often exhibit collective transitions emerging from interactions across many components. Classical stability theory describes such transitions in spectral space, where dynamics is organized by spatially extended global eigenmodes whose collective nature obscures direct association with individual physical components. Here, we show that structural disorder in empirical random networks can fundamentally alter this picture. These properties induce spectral localization, causing Laplacian modes to concentrate on small subsets of nodes and producing a mode--node correspondence in which collective dynamics becomes governed predominantly by the local behavior of a dominant node together with their effective coupling to the surrounding network. As a consequence, stability properties can be interpreted directly in node space rather than purely in spectral space. Exploiting this principle, we develop a node-resolved framework that predicts transition onsets, identifies the nodes responsible for emergent collective behavior, and restores interpretability in systems where classical modal theories fail. In heterogeneous reaction networks, the same mechanism gives rise to exotic collective states where different subsets of nodes develop distinct dynamical behaviors beyond those associated with homogeneous assumptions. Our results show that complex network structures naturally generate spectral localization, revealing the microscopic drivers of macroscopic dynamics.
In the framework of the one-dimensional nonlinear Schrodinger equation (1D-NLSE) of the focusing type, the present paper studies numerically rogue waves (RWs) that emerge in the nonlinear stage of noise-induced modulational instability of a plane wave. For a large ensemble of simulations, all sufficiently large local maximums of the wavefield (i.e., RWs) are systematically collected and analyzed. It is shown that the frequency of occurrence of such maximums begins to increase from zero at the time, when the fourth-order moment of amplitude reaches its first (largest) local maximum, and then grows in an oscillatory manner approaching its asymptotic value at long time. Near the statistically stationary state, the 1D-NLSE generates a much larger number of RWs than a comparable linear system, but, in average, one RW affects a much smaller spatiotemporal area. The distribution of these RWs by maximum intensity represents an exponential-like function without noticeable changes in behavior, indicating a similar origin for RWs of significantly different amplitudes. The collected RWs are compared within a sufficiently large spatiotemporal window with nine exact solutions, of which two models reproduce RW dynamics much better than the others: a general collision of two solitons and a general collision of two Tajiri-Watanabe breathers; RWs are fitted with these collisions using pre-compiled databases of such solutions. The parameters of the two models turn out to be quite similar to the direct scattering transform spectrum of the whole wavefield, supporting a hypothesis that RWs may emerge due to synchronization of a few nonlinear modes (solitons or breathers) in presence of many other nonlinear modes of the wavefield.
We extend the ``Brownian bees'' model of Berestycki et al. (2021, 2022) to cooperative reproduction, $kA\to(k{+}1)A$, of a population of $N$ symmetric random walkers with removal, at each birth event, of the particle farthest from the origin. Working in the limit $N\to\infty$, we formulate a hydrodynamic free-boundary problem for this model. Using this formalism, we determine steady state population densities for all~$k$ and prove their linear stability for $k\le 2$ and instability for $k\ge 4$. In the marginal case $k=3$, there is a whole continuous family of steady states at a single, critical ratio of the reproduction and diffusion rates. Above criticality the population undergoes an asymptotically self-similar finite-time collapse to the origin. Below the criticality the population spreads diffusively, but the reproduction remains quantitatively relevant. For $k\ge 4$, the unstable steady state separates regimes of a finite-time collapse and a diffusive spreading. Here the collapse dynamics is asymptotically self-similar, and the population density exhibits a scale separation requiring a matched-asymptotic description. Our analytical predictions are confirmed by numerical solutions of the hydrodynamic free-boundary problem and by Monte Carlo simulations of the original microscopic model.
Acoustic horizons in Bose--Einstein condensates are usually characterized through long-wavelength Bogoliubov phonons. We study a nonlinear counterpart: whether a one-dimensional dark-soliton branch can sustain upstream laboratory motion in a stationary transonic flow. The mechanism is local at leading order. A regular dark soliton has a bounded fluid-frame velocity, limited by the local sound speed; therefore, in the supersonic region, the background flow exceeds the largest upstream velocity available to the soliton branch. The sonic point is thus the upstream edge of the local dark-soliton branch, rather than a hard wall or a soliton geodesic surface. We construct stationary transonic Gross--Pitaevskii backgrounds and evolve the full order parameter in an open, nonperiodic domain. The simulations show upstream propagation on the subsonic side, finite-depth stalling on the subsonic side, and downstream advection for defects initialized in the supersonic region with upstream velocity relative to the fluid. Convergence, branch-consistency, local-density, phase-jump, and a dense scan of dimensionless upstream attempts support the soliton-like interpretation. The result is a branch-existence constraint, not a rigorous lower bound on arbitrary density minima of the Gross--Pitaevskii field.
We present a field-theoretic construction of a class of artificial neural networks from solitonic degrees of freedom in nonlinear scalar field theory. The purpose is not to rename a standard neural layer in the language of solitons, but to start from a continuum action, restrict the theory to a nonperturbative sector containing localized stable solutions, perform a collective-coordinate reduction, and derive the neural layer as the finite-dimensional input-output map of the reduced solitonic dynamics. In this construction, the computational unit is a projected collective coordinate of a localized field configuration rather than an elementary point variable; the activation function is the solitonic response profile or scattering map; the weight matrix is the Hessian or overlap matrix of an effective interaction energy among solitons; the bias is induced by external sources, vacuum asymmetry, or boundary forcing; and depth is a discrete evolution parameter on the solitonic moduli space. We develop the construction explicitly for the \(\phi^4\) kink, where the \(\tanh\) activation and the logistic sigmoid arise from the kink profile, and then derive the multilayer feedforward form from an operator-splitting approximation to collective-coordinate gradient flow. We emphasize the novelty criterion: the neural architecture is obtained only after specifying the field action, the solitonic ansatz, the moduli-space metric, the interaction functional, and the projection map. The result is a controlled route from nonlinear field theory to neural-network structure, with robustness tied to the energetic and topological stability of the solitonic sector.
In a recent work we studied the first nonlinear stage of modulation instability (NLSMI) of x-periodic anomalous (rogue, freak, extreme) waves (AWs) of physically relevant multidimensional (generalizations of the focusing) nonlinear Schr\"odinger (MNLS) equation, like the non integrable elliptic and hyperbolic nonlinear Schr\"odinger (NLS) equations in d+1 dimensions, d = 2, 3, in the quasi one dimensional (Q1D) regime in which the wavelength in the direction of propagation x is small with respect to the wavelengths in the transversal directions. We showed that, at leading order, the first NLSMI is universal, independent of the particular MNLS model, and described by suitable adiabatic deformations of the quasi-homoclinic Akhmediev breather solution of NLS, in excellent agreement with numerical simulations. In the present work we focus on the recurrence of x-periodic AWs in the Q1D regime. We show that, although the first nonlinear stage of MI is essentially universal for all MNLS equations, the recurrence dynamics exhibit significant O(1) differences among different models. Moreover, successive nonlinear stages generally display increasingly complex combinations of fission and fusion processes, leading to progressively richer dynamical choreographies. Since MNLS equations in the Q1D regime can be viewed as multidimensional perturbations of the integrable NLS equation, we use the recently developed finite gap perturbation theory of NLS AWs to give an analytic and quantitative description of the recurrence of Q1D AWs, in excellent agreement with numerical simulations. Due to the physical relevance of the MNLS equations considered in this work, and due to the universality of the processes discussed in this paper, it is plausible that they be observable in many fields of physics, like water waves, nonlinear optics, plasma physics, Bose-Einstein condensates, etc . . .
We investigate the dynamical mechanisms underlying the contrasting Floquet spectral evolutions observed in viscously damped and in nonlinear mean-flow damped higher-order nonlinear Schr\"odinger models. Motivated by the persistent organized Floquet-band structure under nonlinear mean-flow damping and the repeated Floquet band reconnection observed under viscous damping, we derive a reduced five-mode carrier-sideband truncation and reformulate the dynamics in amplitude-phase variables to isolate the dominant interaction phases associated with the principal four-wave interaction products. Within this framework, viscous damping acts primarily modewise and does not directly modify the leading interaction-phase dynamics. By contrast, nonlinear mean-flow damping contributes directly to the interaction-phase evolution through interaction-dependent dissipative corrections. In the carrier-sideband regime, these corrections generate terms of the form $-\kappa_j \sin(\psi_j)$, introducing dissipative feedback into the dominant carrier-sideband interaction dynamics. To interpret the resulting interaction-phase evolution, we examine recurrent finite-gap NLS benchmark solutions whose modulation dynamics are independently understood. These benchmarks show that substantial interaction-phase evolution and localized restructuring may occur even within organized quasiperiodic dynamics possessing invariant Floquet spectral structure. Numerical diagnostics show that the nonlinear mean-flow damped system exhibits persistent recurrent carrier-sideband focusing dynamics together with organized Floquet evolution despite substantial interaction-phase restructuring, whereas the viscous system exhibits progressively diffuse modulation dynamics together with repeated Floquet reconfiguration and weakening persistence of the recurrent carrier-sideband interaction structure.
We theoretically investigate the effects of parametric driving on the one-dimensional Frenkel-Kontorova model, a nonlinear many-body lattice system. It is numerically found that a parametric vibration induces spatiotemporal ordering characterized by the subharmonic frequency $\omega_{\mathrm{ex}}/2$ and wavenumber $k$ which takes nontrivial values depending on the vibration strength $P_{\mathrm{ac}}$ and frequency $\omega_{\mathrm{ex}}$. With increasing $P_{\mathrm{ac}}$, $k$ gradually increases and discontinuously jumps up to $k=\pi$. Based on an extended linear stability analysis, we show that the former $k$ resonance (the latter $\pi$ resonance) corresponds to the unstable (stable) mode and that the $k$ resonance is made possible by the interplay between the parametric amplification and the collective fluctuation developing through the nonlinear effect.
Three-wave interactions (or resonant triads) are the lowest-order nonlinear interaction in pattern formation and arise between waves with different orientations when the sum of two wavevectors equals a third one. When a pattern has only one length scale, stripe patterns are possible but three-wave interactions are responsible for the prevalence of hexagons close to onset. In problems with two length scales, there is a much wider range of possible three-wave interactions, leading to more complex structures such as superhexagons, stars, quasipatterns and even spatiotemporal chaos (STC). We investigate the role that nonlinear wave interactions play in the formation of STC in a model partial differential equation (PDE) in the case that the length scale ratio is $\sqrt{7}$, relevant to superlattice patterns in the Faraday wave experiment. The simpler aspects of the dynamics can be represented by a system of ordinary differential equations (ODEs) derived from the PDE using weakly nonlinear theory. We analyze the equilibrium patterns in these ODEs and evaluate their stability, comparing the results with direct numerical simulations of the model PDE. The ODEs predict parameter regimes where there are no stable simple equilibria, which is where we typically find complex behavior in the PDE. We have conducted a careful study of the transition from simple patterns to patterns that include modes beyond the finite-dimensional subspace imposed in the reduction to the ODEs, to time-dependent competition between different triads, ending up with fully developed STC. For our choice of length scale ratio, we show that four-wave interactions also play an important role. Our analysis is relevant to any pattern-forming system with three-wave interactions involving two length scales, such as the Faraday wave experiment, coupled reaction-diffusion systems, and pattern formation in dryland vegetation.
We investigate the emergence of chaotic dynamics in collective-coordinate reductions of a driven and spatially modulated $\phi^4$ field describing the motion of topological kinks. Focusing on finite-dimensional effective models, we consider both translation-only and constraint-consistent two--collective--coordinate reductions in the presence of spatial pinning, dissipation, and traveling-wave forcing. Using Melnikov theory, we obtain an explicit analytical characterization of separatrix splitting and derive closed-form criteria for the onset of chaotic dynamics in the reduced phase space. In the two--collective--coordinate framework the Melnikov analysis is formulated in an extended phase space, allowing the distinct roles of translational motion and internal-mode excitation to be identified. Numerical simulations of the reduced systems, including stroboscopic Poincar\'e sections and Lyapunov exponent computations, confirm the analytical predictions and reveal chaotic layers organized around the unperturbed separatrix.
We study of the formation of pattern-forming fronts in the presence of a rigidly-propagating parameter ramp which is slowly-varying in space. In the context of the prototypical supercritical complex Ginzburg-Landau equation, we show that not only the leading order front interface, but also the selected spatial wave number is governed by the transition of the ramp between absolute and convective instability. The slow ramp then induces a further delay of the front interface and perturbation of the selected wave number, controlled by the slow passage near a complex fold of strong- and weak-stable eigenspaces. To analyze the behavior near this fold, we perform a multiple scales analysis to predict the higher-order front interface delay in terms of zeros and poles of a complex Airy quotient inner solution. We confirm these predictions with numerical continuation of heteroclinics in the associated traveling wave equation. We also numerically characterize their spectral stability, finding accumulation of eigenvalues consistent with previous results on slow absolute spectrum. We then show the leading-order absolute/convective instability heuristic accurately describes selected wave numbers in an analogous slowly-ramped Swift-Hohenberg equation.
Microresonator frequency combs are key components for integrating optical devices into photonic circuits. They provide high stability, coherence, and low noise, even without external stabilization. Yet microcomb design remains largely heuristic: waveguide and resonator parameters are typically swept manually or semi-empirically, and the resulting spectra are evaluated only afterwards. This forward-design workflow is computationally costly, relies heavily on designer intuition, and does not generally identify optimal solutions. Here, we present an adjoint-based inverse-design framework for microresonator frequency combs that directly optimizes the comb spectrum with respect to pre-defined objectives. We demonstrate the power and flexibility of this approach by addressing three challenging problems: designing spectrally flat combs, synthesizing arbitrarily shaped comb spectra, and enforcing several performance metrics simultaneously through multi-objective optimization. Our results show that inverse design offers a systematic and efficient route to compact on-chip light sources with properties tailored to diverse applications.
We study the nonlinear wave dynamics of one-dimensional chains of polycatenated rings. These interlocked structures support amplitude-dependent nonlinear wave propagation driven by tensile activation and internal structural flexibility, unlike traditional granular crystals. Through dynamic impact experiments, finite-element modeling, and discrete-particle simulations of vertical chains pretensioned by gravity, we observe and explain nonlinear waves characterized by a compact leading wavefront followed by persistent trailing oscillations, which arise from energy partitioning into the rings' internal bending modes. Further, we demonstrate that the system's nonlinearity is not a fixed material constant. By altering the rings' geometric aspect ratio and contact angles, we can tune the effective contact exponent and the amplitude scaling of the wave speed. This work builds upon nonlinear wave propagation in classical granular crystals and establishes polycatenated systems as a highly tunable and designable platform to study and control nonlinear dynamics.
We prove a no-go theorem for the mass-radius relation of localized and stable field configurations, known as solitons. Defining the mass-radius index by $\Gamma \equiv \frac{{\rm{d}}\ln M}{{\rm{d}}\ln R}$, for real scalar field theories in $d$ spatial dimensions, we show that typical non-topological, non-relativistic, and spherically symmetric solitons cannot have $\Gamma$ in the range $[0, d]$. The forces considered originate from gradient energy, self-interaction, and gravitation, with the typicality assumption excluding the fine-tuned region of the parameter space where all three forces have comparable strength. Importantly, the theorem works for an arbitrary self-interaction that, in the relativistic theory, is allowed to be non-power-law in the field, be non-analytic around the classical vacuum (where the field amplitude vanishes), or to include derivative couplings. Additionally, the theorem makes no assumptions about the explicit form of the soliton's density profile or the behavior of $\Gamma$ as a function of $R$. We also argue that the same exclusion applies to compact objects formed from self-gravitating, non-relativistic, barotropic fluids with arbitrary equations of state. As a consequence for cosmology, it is worth noting that observations favor a core in the centers of dark matter halos with $\Gamma \simeq 1.7$, which (for $d=3$) lies approximately in the middle of the excluded range. Therefore, proposals such as ultra-light or fluid-like dark matter models are essentially ruled out as natural explanations for halo cores, provided other astrophysical effects are negligible.
Both the direct method and KdV approximation match simulations for rarefaction and dispersive waves arising from Riemann data.
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In the present work, we revisit the Adlam-Allen (AA) model in order to investigate its numerically observed rarefaction and dispersive shock waves that arise in numerical simulations of the Riemann problem associated with the model. On the one hand, we perform a direct analysis of the rarefaction and dispersive shock waves of the AA model via examining its corresponding dispersionless system and leveraging the DSW-fitting method to obtain theoretical predictions on various edge features of the dispersive shock waves. On the other hand, we review the KdV reduction of the AA model and utilize the KdV dispersive shock wave to approximate that of the AA model. Relevant numerical comparisons demonstrate the good performance of not only the direct analysis on the AA dispersive shock wave, but also of the approximation via the KdV DSW. These methodologies provide a systematic toolbox for analyzing the outcome of Riemann problems in not only this fundamental setting of cold plasmas but also potentially in related plasma-physics problems.
In the present work we consider the subject of dark fractional solitary waves in the realm of generalized (fractional) forms of the nonlinear Schr\"odinger (NLS) equation. While earlier studies have examined such states in the realm of real field theories, we showcase the existence and stability of individual dark solitary waves in such NLS settings and subsequently turn to two-soliton solutions. We find different branches of such two-soliton solution equilibria and contrary to the real field-theoretic setting all possible branches of two-soliton equilibria are found to be potentially unstable, although with different types of instabilities. Odd branches are potentially subject to oscillatory instabilities, while even branches are always exponentially unstable. The dynamics that results from the instabilities is also examined and is found to potentially feature breathing characteristics. This prompts us to seek and find associated periodic (breathing) orbits that are also unprecedented in this context, to the best of our knowledge. The effective particle-like dynamics of the solitary waves also prompts us to seek ordinary differential equation (ODE) descriptions to the dark soliton interaction dynamics. These are shown to hold promise toward providing us with effective reduced order models, indicating some potential directions for further investigation.
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.
Discrete nonlinear systems support a rich variety of localized and extended wave phenomena, with their dynamics sensitively dependent on the symmetries of the underlying interaction forces within the lattice. Odd elasticity, emerging in effective models of active materials, breaks the action-reaction symmetry of the local interactions and gives rise to new wave behavior. We investigate the existence and stability of traveling waves in a nonlinear lattice with odd elasticity, where the coupling force between adjacent units depends asymmetrically on the deformations of the coupled units (nonreciprocal elastic coupling). We demonstrate the existence of periodic and quasiperiodic traveling waves and analyze their spectral stability using the master stability framework. In particular, we identify the onset of Eckhaus instability based on the curvature of the associated master stability curve. This approach enables a quantitative analysis of size effects, specifically the bounds on lattice sizes for which a given traveling wave is stable. The stability analysis for quasiperiodic waves is based on an effective description of the envelope of the response through a rotating wave approximation, which agrees well with direct numerical simulations. Our findings establish a unified framework for understanding wave propagation characteristics in nonlinear lattices, for both periodic and quasiperiodic wave profiles. We highlight, qualitatively and quantitatively, the role of nonreciprocal stiffness on the existence and stability of nonlinear traveling waves in dissipative systems, and discuss how localization and stability depend on the interplay between nonlinearity, dissipation and odd elasticity.
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.
Transition waves are common in multistable mechanical metamaterials, and the dynamics of weakly discrete transition waves under driving forces have been extensively discussed. However, as lattice effects become more pronounced, strongly discrete transition waves may exhibit dynamics that cannot be predicted by the continuum limit. Here, by tilting a bistable chain, we introduce a gravitational perturbation term into the dynamical equations, under which the transition waves are continuously accelerated. In the strongly discrete regime, we find that transition waves under gravitational driving possess quasi-stationary velocity plateaus (QSVPs), and the number of these plateaus first increases and then decreases as the tilt angle increases. We theoretically elucidate that the emergence of the velocity plateaus originates from the balance between gravitational driving and phonon radiation. In further analysis, the theoretical model reveals that the balance point undergoes a bifurcation at the radiation resonance, which leads to a change in the number of velocity plateaus. Our study extends the investigation of transition waves into the strongly discrete regime, and the emergence of multiple velocity plateaus opens up new possibilities for programmable solitary waves.
We study the ground-state density patterns of a population-imbalanced two-component dipolar Bose-Einstein condensate confined in a circular quasi-two-dimensional box. Using a mean-field model, we map out phase diagrams as functions of the axial confinement, interaction imbalance, and population ratio. The system supports a rich sequence of stationary morphologies, including a nearly uniform pancake state, pancake-droplet and ring-droplet coexistence states, droplet arrays, and concentric rings. These patterns show a close structural correspondence to microphase-separated morphologies in diblock-copolymer systems, with the population imbalance acting as an effective volume fraction that selects the pattern topology. Analysis of the density profiles and structure factors reveals that the modulated states possess an intrinsic nonzero characteristic wave vector, which remains essentially unchanged when the box size is varied. We also find that the characteristic pattern spacing scales linearly with the axial confinement length, indicating that the transverse thickness of the condensate controls the effective in-plane length scale. In a finite circular box, this smooth scaling is interrupted by discrete steps, reflecting geometric frustration and the integer locking of the number of rings or droplets. Our results show that box-trapped dipolar mixtures provide a controllable platform for studying finite-size pattern selection and nonlocal microphase formation in quantum fluids.
We study an agent-based model of self-propelled particles with a velocity-dependent alignment rule. This interaction is orientation weighted and acts along the line connecting neighboring particles. Tuning the alignment strength produces several distinct collective regimes, including disordered gas-like motion, coherent flocking, jammed high-density states, and densely ordered moving clusters with active-crystal-like behavior. These results show that a simple local alignment rule can generate a broad range of nonequilibrium collective dynamics within a single microscopic model.
We investigate the propagation of fronts in an inhomogeneous medium within the framework of the $\phi^4$ model. The inhomogeneity is modeled either as an interface separating regions with different dissipation or as a finite layer with modified dissipation. The propagating front is described in two ways: as a kink solution in an effectively unbounded domain, and as a half-kink in a finite system. The half-kink represents the decay of the unstable state $\phi=0$ toward the true vacuum. We show that while the effective description based on the kink provides accurate results, applying a similar approach to the half-kink leads to significant deviations from the predictions of the field model. We then demonstrate that a consistent description does exist and propose a modified effective model which reproduces the field-theory results over a relatively broad range of parameters.
We investigate the robustness of Constantin's explicit reconstruction formula for two-dimensional irrotational solitary water waves. This formula recovers the free-surface profile from the dynamic pressure trace at the bed and depends on both the wave speed and the undisturbed depth. We consider simultaneous perturbations in these three quantities and derive an $L^2$ error estimate for the reconstructed profile. The proof uses the hodograph transform, holomorphic extension arguments, and Paley--Wiener Fourier-decay estimates, yielding stability estimates with sublinear dependence on the perturbation size. We include numerical computations to illustrate the effects of specifically designed perturbations.
We investigate how localized inhomogeneity affects the geometry and stability of migratory bird formations. We use a lifting-line model with a horseshoe-vortex representation to describe the longitudinal dynamics of aerodynamic interactions. As a reference case, we first analyze homogeneous formations and show that their steady states exhibit a U-shaped geometry with hierarchical streamwise spacing, in which adjacent birds become progressively closer toward the leader. We then introduce localized inhomogeneity by modifying the wingspan of a single bird, with its physical properties determined by scaling relations. We determine the range of wingspan variation that preserves a stable formation. The stability range depends strongly on the position of the modified bird, being narrower near the outer wing and broader near the leader. These findings provide a minimal dynamical framework for understanding how local aerodynamic interactions and localized individual differences affect collective flight structures.
The extended KdV equation is a nonlinear dispersive wave model that is asymptotically or variationally derived from the full dispersive Euler shallow water waves equations when gravity-capillary and higher order nonlinear effects are taken into account, under weakly nonlinear and long-wave approximations. This reduction introduces four additional terms beyond the classical KdV equation: a nonlinear term (quadratic nonlinearity), two nonlinear-dispersive terms, and a fully dispersive term (fifth order dispersion). In this paper, we employ a variational approach based on averaged Lagrangians to analyze the accuracy of single solitary wave solutions governed by a particular extended KdV equation where energy conservation is a key feature. Compared with solitary wave solutions previously obtained through higher order asymptotics and algebraic methods, the present variational solutions are notably simpler and more readily applicable to practical problems. The solitary wave solutions obtained through this method are then systematically compared with direct numerical simulations, and the corresponding results are critically discussed. We further demonstrate the applicability of these single solitary waves to problems in the field of non-convex dispersive hydrodynamics. These problems include shallow water classical undular bores, commonly known as dispersive shock waves, and non-classical (resonant) dispersive shocks which are additionally analyzed using the concept of Whitham shocks. Theoretical predictions show excellent agreement with numerical simulations.
A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi-Pasta-Ulam lattice with a general cubic interaction force. Explicit periodic traveling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink, and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic traveling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic speeds is shown to lead to modulational instability of periodic traveling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic traveling waves and initial value problems that also reveal additional short-wave instabilities.
We study the collision dynamics of localized oscillons in two classes of $(1+1)$-dimensional scalar field theories with metastable false vacua, a normal class with a positive quartic self-interaction term and an inverted class with a negative quartic term. We construct small-amplitude oscillon solutions around the false vacuum using the Fodor {\emph{et al.}} expansion, and show that the force between oscillons decays exponentially at large separation, with a strength modulated by their relative phase. Numerical simulations of two-oscillon collisions exhibit reflection, crossing, and formation of excited oscillons. Resonance windows occur, similar to those found in kink-antikink collisions. In the normal theory, if the oscillons have sufficient energy, the field can pass over a sphaleron barrier and evolve into a kink-antikink pair, initiating a phase transition to the true vacuum. We also simulate the collision of oscillons evolved from a slightly perturbed sphaleron.
Localized patterns are spatially confined structures that arise in lattice dynamical systems and play an important role in physics, biology, and materials science. While their existence and bifurcation structure are well-understood, the stability of these solutions remains largely unexplored, particularly in discrete and high-dimensional settings. In this work, we develop a general theoretical framework to analyze the spectral stability of localized steady states in one-dimensional and multi-dimensional rectangular lattices. Our approach leverages the properties of front and back solutions, combined with a discrete Evans function formulation, to characterize the spectrum of localized solutions. We prove that, for well-separated regions of localization, the Evans function asymptotically factorizes into contributions from the underlying fronts and backs, allowing explicit counting of unstable eigenvalues. This framework applies to solutions with single or multiple plateaus, including oscillatory and multi-pulse configurations. We illustrate the results on a real-valued cubic-quintic Ginzburg-Landau lattice, a prototypical Nagumo-type system, and provide numerical demonstrations of bifurcation structures and eigenvalue spectra.
Properties of spatially dependent relaxation oscillations near a SNIPER bifurcation are described. A SNIPER bifurcation creates a large-amplitude long-period periodic orbit via the annihilation of a pair of fixed points in a saddle-node bifurcation. We show that in spatially extended media, this orbit may undergo a long-wavelength instability, leading to spatially modulated oscillations that persist on both sides of the SNIPER. The oscillations take different forms depending on the system: a chimera state in a theta-reaction-diffusion model, and chaotic spiking in an activator-inhibitor-substrate model. The results are expected to have applications in a number of physical systems exhibiting SNIPER bifurcations, ranging from models of the nervous system through chemical reactions to nonlinear optics.
We experimentally generate sine-Gordon-like solitons in a spin-1 spinor Bose-Einstein condensate (BEC) utilizing a robust and reproducible local phase-imprinting scheme. We find that the soliton velocity can be tuned by the effective quadratic Zeeman shift. This enables the investigation of controlled soliton interactions, in which we observe the characteristic elastic collision behavior of the integrable sine-Gordon model. The spatial displacement -- the so-called phase shift -- between incoming and outgoing solitons, the signature of their pairwise interaction, is found to be in quantitative agreement with numerical spin-1 simulations within the error bars. These results establish spinor BECs as a highly controllable experimental platform for studying aspects of the dynamics of sine-Gordon-like models.
Cyclops states - three-cluster configurations consisting of two synchronous groups and a solitary oscillator - dominate in ensembles of phase oscillators with inertia and multiple coupling harmonics [Phys. Rev. E 109, 054202 (2024)]. In this work, for the first time, we systematically study nonstationary cyclops states that preserve the three-cluster structure: breathing and rotobreathing cyclops states. We identify two scenarios for their destabilization: period doubling, leading to quasicyclops states while preserving frequency synchronization within the clusters, and the destruction of one or two clusters, resulting in the emergence of switching cyclops or multicluster states. We show that breathing and rotobreathing cyclops states occupy vast parameter regions of the second coupling harmonic and are key elements of the dynamics. The results are important for predicting and controlling complex collective states in ensembles with higher-order interaction harmonics of various natures.
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