Papers

We introduce a procedure to obtain lump solutions via the formation of a kink-antikink pair, consisting of the superposition of kinks whose distance from the origin is controlled by a single parameter $a$. For large values of $a$, a wide plateau appears in the solution, which we call a large lump. The procedure involves the use of a first-order equation that allows the construction of the potential associated with the lump solution. We then investigate several known scalar field models where the parent kinks are capable of giving rise to novel lumps. The lump inherits the tails of the parent kink, allowing for either short-range exponential profiles or long-range profiles characterized by distinct power-law decays. We also show how to verify if an arbitrary lump solution can be obtained via our method and illustrate this possibility with a novel vacuumless lump.

A one-dimensional SQUID (superconducting quantum interference device) array/metamaterial is investigated numerically with respect to its localization properties due to nonlinearity in the absence of dissipation and periodic driving. The system possesses a conserved Hamiltonian function representing its energy, and supports localized modes of the discrete breather type even in the presence of a moderately high dc flux bias. The appearance of discrete breathers in that system has been largely overlooked in literature. We find a new type of discrete breather that is asynchronous, meaning that the frequency of oscillation of the SQUID at the central breather site is different than that of the SQUIDs at the other sites of the metamaterial. Nonlinear localization is investigated by initializing the system with a single-site excitation of given amplitude (initial amplitude) for a fixed value of the coupling coefficient, while parameters such as the dc flux bias, the single-site initial excitation amplitude, and/or the SQUID can vary independently. Using the energetic participation ratio as a measure of the degree of localization, the existence of asynchronous highly localized modes and transitions between delocalized (extended) and localized modes re identified.

We study a two–dimensional discrete nonlinear Schrödinger equation with saturable nonlinearity on the lattice $\mathbb Z^2$. Using a variational approach based on the Nehari manifold, we establish the existence of nontrivial periodic ground states on finite lattices and establish the existence of exponentially localized ground states in $\ell^2(\mathbb Z^2)$. A principal result is the rigorous passage from periodic to localized states: we show that, up to lattice translations, periodic ground states converge strongly in $\ell^2(\mathbb Z^2)$ to a localized ground state as the lattice periods tend to infinity. The analysis combines variational methods, spectral properties of the discrete Laplacian, and concentration–compactness techniques adapted to the two–dimensional discrete setting. We further derive qualitative properties of the resulting solutions, including positivity and exponential localization, and establish a conditional orbital stability result within the Grillakis–Shatah–Strauss framework. Numerical computations illustrate the theoretical results and confirm the predicted convergence and localization behavior.

We develop a systematic adiabatic perturbation theory for the integrable $F=1$ spinor nonlinear Schrödinger equation under nonvanishing boundary conditions, formulated entirely within the framework of the associated Riemann–Hilbert problem. In this setting, localized nonlinear excitations are characterized by discrete spectral data consisting of a complex eigenvalue and an associated polarization vector. For a general class of small perturbations preserving the background, we derive the perturbation-induced evolution of the scattering data directly at the level of the Riemann–Hilbert problem. In the one-soliton sector, this yields a closed finite-dimensional dynamical system governing the slow evolution of the effective soliton parameters, including the spectral variables, the soliton center and phase, the residue amplitude, and the internal polarization state. The latter evolves according to a constrained dynamical equation with no scalar analogue. For localized perturbations, the modulation equations are expressed in explicit integral form in terms of the one-soliton eigenfunctions, providing a fully computable description of the dynamics. In the limit of vanishing boundary conditions, the resulting system reduces to the perturbation theory obtained by E. V. Doktorov, et al, Phys. Rev. A 77 (2008), no. 4, 043617.

We investigate the dynamics and spectral stability of traveling kink and antikink solutions in a dissipative sine–Gordon equation with two distinct fourth–order regularization mechanisms: a mixed space–time (inertial) term and a purely spatial (elliptic) term. The model includes damping, bias forcing, and higher–order dissipative effects, and is motivated by refined descriptions of fluxon dynamics in long Josephson junctions. Using a collective–coordinate reduction, we derive a Melnikov–type condition for speed selection, yielding explicit predictions for asymptotic propagation speeds, which are validated by direct numerical simulations of the full partial differential equation. Spectral stability is analyzed using Evans function techniques adapted to the singular slow–fast structure induced by the regularization. By formulating the linearized problem on a consistent–splitting domain, we show that no additional point spectrum bifurcates from the origin. Near the edges of the essential spectrum, a square–root transformation is used to resolve branch singularities and establish analyticity in a lifted spectral variable. Numerical Evans function computations near $\lambda=0$ and near the essential spectrum edges confirm the analytical results, indicating absence of unstable eigenvalues for both kink and antikink solutions.

Self-organized spatial patterns are central to nonequilibrium physics and cell biology, yet locating instabilities in multi-component, reaction-diffusion networks remains challenging because standard eigenvalue analyses scale with the number of biochemical states and rely on reaction kinetics often poorly constrained by experiments. Exploiting the common mass-conserving structure of protein reaction kinetics and the fact that nonlinear feedback is typically confined to membrane reactions while lateral membrane diffusion is negligible, we develop a geometric classification that predicts stationary, and approximately also oscillatory, pattern-forming instabilities from reactive equilibria. The developed criteria reduce the stability analysis from the full component space to the space of conserved species. On this reduced space, slope matrices—describing the change of equilibrium cytosolic densities with respect to total species densities—govern onset. Based on densities in chemical equilibrium, this approach eliminates the requirement of comprehensive kinetic knowledge frequently lacking in experimental systems. Thus, a broad range of instabilities can be understood as mass-redistribution instabilities—self-amplifying mass redistribution caused by shifting local equilibria. We apply these criteria to models for the Escherichia coli Min system and the Caenorhabditis elegans polarity system and show that the reduction extends to the mixed-dimensional dynamics in systems coupling bulk cytosolic dynamics with membrane dynamics on the boundary. Together, these results provide an interpretable, broadly applicable, and experimentally accessible framework for diagnosing and designing pattern formation in multicomponent nonequilibrium systems on the basis of conservation laws.

Maxwell's wave equation in the presence of a cubic nonlinearity and a periodically time-varying refractive index (a photonic time crystal) is reduced, for spatially monochromatic waves, to a nonlinear Mathieu equation. Near the principal momentum gap this equation admits an autonomous two-dimensional reduction whose complete Hamiltonian phase portrait can be obtained analytically. We derive the two homoclinic separatrices corresponding to temporally localised momentum gap solitons, identify the nonlinear centres and the critical Hamiltonian value $H_c$, and calculate the point of maximum linear parametric gain. We then consider spatially localised pulses and show how the nucleation of multiple spatiotemporal gap solitons can produce a broad supercontinuum in momentum space; for stronger seeds, transient extreme nonlinear localisation can accompany an abrupt additional broadening of this momentum spectrum. These results establish a direct connection between Floquet amplification, nonlinear saturation, homoclinic dynamics, and momentum space spectral broadening in nonlinear photonic time crystals.

Nonequilibrium membrane pattern formation is studied using meshless membrane simulation. Bound proteins are considered to have two states that generate different membrane spontaneous curvatures. Protein binding and unbinding occur cyclically owing to chemical potential differences, as an off-lattice active Potts model. It is found that this cyclic binding/unbinding can induce blinking domains, with oscillating size: convex domains of the proteins with a higher spontaneous curvature grow, and subsequently, the proteins change to the other state with a lower spontaneous curvature, resulting in domain shrinkage. These processes repeat. In thermal equilibrium, hexagonal convex domains are formed by the competition between bending and surface tension energies, so that they are stably formed only under positive surface tension. However, blinking domains can form even in tensionless membranes.

In this paper, we travel through an unknown territory of dromions, to unearth and show the new properties/attributes of dromions which have never been brought to the fore since they were first discovered by Boiti etal [1]. These new attributes are brought out by investigating a generalized (2+1) dimensional Nonlinear Schrodinger (NLS) equation by exploiting Truncated Painleve approach. The signatures that can be attributed to dromions include existence of firewall and reflection at the boundary, uneven distribution of energy among different bound states, amplitude dependence on adjacent dromions, etc. We have corroborated the main analytical results with numerical simulations also. We categorically state that these properties are universal and can be extracted in any (2+1) dimensional nonlinear partial differential equation. We do believe that these properties which shed more light on the behaviour of dromions may have wider repercussions in nonlinear optics, Bose-Einstein condensates and plasma physics.

In this article, the nonlinear Helmholtz equation with non-Kerr nonlinearity, such as self steepening and self frequency shift, is considered. A traveling wave transformation is applied, and the extended nonlinear Helmholtz equation is reduced to a Hamiltonian dynamical system. Then, the reduced Hamiltonian system is analyzed by classification of equilibrium points, phase space analysis, and the construction of exact wave solutions. The relationship between the reduced dynamical coefficients and the original physical parameters is further established through a parameter space analysis. It is shown that self steepening directly modifies the reduced dynamics, whereas self frequency shift acts through the compatibility condition for the real traveling wave reduction. Together, these non-Kerr effects reshape the phase space geometry and traveling wave structure. Localized and periodic traveling waves are obtained, with their existence determined by the balance among dispersion, nonparaxiality, Kerr nonlinearity, and non-Kerr effects. Furthermore, a periodically forced version of the reduced system is examined to study the transition from regular to irregular dynamics. It has been observed that external forcing can induce complex oscillatory behavior. Bifurcation analysis, time series evolution, phase space analysis, largest Lyapunov exponent, and Poincaré section demonstrate the emergence of quasiperiodic and chaotic responses under sufficiently strong forcing. All analytical branches are verified through full-equation residual evaluation, while a few selected branches are additionally examined through direct numerical propagation and robustness tests under complex Gaussian perturbations. The results show that self steepening directly renormalizes the effective nonlinear dynamics, whereas self frequency shift restricts the admissible real-envelope traveling wave manifold.

We present quasistationary ring-shaped soliton necklaces in a two-component envelope propagating in a medium with competing cubic-quintic nonlinearity. Metastable propagation of soliton necklaces results from a balance of repulsion between adjacent out-of-phase solitons in one component and confinement by the boundary of a flattop region in the other. Numerical simulations demonstrate metastable propagation over about a hundred diffraction lengths, even with random noise added to the input envelopes. The maximum number of solitons in metastable necklaces can be controlled either by changing the size of individual solitons or by adjusting the width of the flattop region hosting the necklace.

We investigate the collision dynamics of two bright sphalerons in a (1+1)-dimensional deformed $\phi^6$ scalar field theory with a symmetric potential possessing false vacua. Two one-parameter realizations of the model, referred to as the barrier and well models, are considered and their static and linear instability properties are first reviewed. We then study head-on collisions of boosted sphalerons over a broad range of initial velocities and deformation parameters. The scattering dynamics exhibit a rich variety of final states, including the production of kink-antikink pairs, long-lived oscillons in true and false vacuum, multiple oscillons propagating in false-vacuum regions, and radiative decay. A particularly remarkable outcome is the emergence of a long-lived bubble of the false broken vacuum bounded by a kink-antikink pair, which repeatedly collapses and re-expands before eventually decaying into an oscillon. These results demonstrate that deformed $\phi^6$ theories with false vacua exhibit considerably richer sphaleron dynamics than previously known and provide new insight into the role of unstable localized configurations in nonlinear field theories.

In this paper we present Open Multi-Processing (OpenMP) Fortran 90/95 programs to solve the Gross-Pitaevskii equation for a rotating dipolar Bose-Einstein condensate (BEC) in two and three dimensions, which is a new version of our previous published programs for a dipolar Bose-Einstein condensate without rotation. After the recent experimental study of a rotating dipolar BEC [L. Klaus et al., Nature Phys. 18, 1453 (2022)], the present programs will be useful tools for related theoretical investigation. The algorithm used is the split-step semi-implicit Crank-Nicolson scheme for imaginary- and real-time propagation to obtain stationary states and BEC dynamics, respectively, as in the previous version [L. E. Young-S. et al., Comput. Phys. Commun. 286 (2023) 108669].

While conventional rogue wave statistical models rely on linear or weakly nonlinear descriptions of random seas, we demonstrate experimentally that moderately or strongly nonlinear wave fields can be represented by sparse ensembles of coherent soliton-like packets. These packets exhibit log-normal amplitude distributions together with uniformly distributed phases and peak emergence times. This sparse coherent structure framework naturally leads to an extreme value description in which the probability of exceedance is governed by the tail of the coherent-structure amplitude distribution. The prediction is validated against laboratory hydrodynamic experiments across a variety of unidirectional sea states, showing good agreement with the experimental observations and comparing favourably with conventional statistical prediction models while retaining analytical simplicity. Our results provide a direct physics-based link between sparse coherent structures and rogue wave probabilities, with broader implications for nonlinear wave physics in optics, cold gases, and plasmas.

We study spatial solitons and their interaction dynamics in nonlinear optical media with spatially engineered refractive index and competing cubic-quintic (CQ) nonlinear profiles, using the variational approximation (VA), the hybrid variational approximation (HVA), and direct numerical simulations. The model was implemented in a symmetric step-index planar dielectric waveguide with a core exhibiting competing CQ nonlinearity and cladding layers possessing only a cubic nonlinear response. For stationary states, the Gaussian VA predicts two types of $N(\mu)$ characteristic curves, where $N$ is soliton norm and $\mu$ is propagation constant, separated by a boundary surface in parameter space, and numerical calculations reveal the same two types. Below this surface, the VA agrees with the numerical results mainly at low powers, while pronounced deviations in the profile and stability appear at high powers. Above the surface, the variational and numerical curves retain the same qualitative form, and a super-Gaussian ansatz accurately describes the high-power flat-top solitons. Soliton collisions produce four post-interaction regimes: Oscillation, Molecular, Splitting, and Breakup. The HVA reproduces the first three over a broad power range, including collisions involving flat-top solitons. Its main limitation arises in the Breakup regime, where strong radiation leaves the guiding region and cannot be represented by the adopted HVA ansatz. Nevertheless, for solitons associated with the second type of characteristic curves, the HVA still captures the breakup dynamics qualitatively. Thus, the HVA provides an efficient description of complex soliton interactions at a substantially lower computational cost than direct numerical simulations.

Four-wave mixing provides a simple setting in which dispersion, nonlinearity, and non-Hermiticity compete to select resonant energy-transfer channels. We study degenerate four-wave mixing in a Kerr dual-core coupler with balanced gain and loss and frequency-dependent intercore coupling. The dispersive coupling changes the symmetry from conventional $\mathcal{PT}$ symmetry to a combined $\mathcal{CPT}$ symmetry and reshapes the two-branch linear spectrum. We determine the unbroken-$\mathcal{CPT}$ domain and classify the branch configurations that can satisfy the degenerate phase-matching condition. In the parameter ranges examined, three resonant channels persist over broad regions, whereas a same-branch channel appears only close to the symmetry-breaking threshold. In this near-threshold regime, a single pump can simultaneously satisfy two distinct nonzero sideband resonances. Direct pulse simulations confirm the predicted resonances and reveal secondary-wave generation and multifrequency cascades near eigenmode coalescence. A reduced three-wave model captures the initial dynamics away from the exceptional point but loses accuracy as the modal basis becomes ill-conditioned. These results show how dispersive coupling reorganizes resonances, group-velocity mismatch, and nonlinear energy exchange in a non-Hermitian wave system, and they identify the exceptional-point region as a regime where a few-mode description can break down.

We investigate two parity-time-symmetric ring resonators coupled over a finite angular region described by a super-Gaussian profile. In the linear regime, analytical spectra are obtained in the homogeneous-coupling and fixed-amplitude narrow-contact limits, while the finite-width problem is treated numerically. Local coupling introduces nonzero spatial Fourier components that mix angular harmonics and lift the degeneracy of counterpropagating modes, resolving each excited doublet into parity-dependent branches. Collisions among these branches generate multiple exceptional-point boundaries and disconnected broken-PT domains. The resulting phase diagrams exhibit re-entrant unbroken-broken-unbroken transitions when the gain-loss strength, coupling width, or peak coupling amplitude is varied. The numerical spectra continuously recover both analytical limits. In the nonlinear regime, selected ground and excited linear modes are used as seeds for adiabatic propagation into finite-amplitude Kerr waveforms that remain dynamically persistent over the simulated observation interval for finite ranges of nonlinear strength. These results show that the spatial profile of inter-resonator coupling provides a geometric means of controlling multimode PT transitions and selecting dynamically accessible nonlinear waveforms in coupled-ring systems.

Two identical scalar dynamical systems coupled through a scalar diffusion equation are studied herein, with respect to bifurcations from a symmetric steady-state to symmetric and asymmetric steady-states and to in-phase and anti-phase oscillations. Numerical continuations based on the developed theory show the shape of the bifurcation branches and the attracting nonlinear states far from bifurcation onset and confirm their, for the most part, derived stability. This study extends the work on the dynamical properties of a single scalar dynamical system coupled to its own delay through an adjacent diffusion field and quantifies further the delay of diffusive information transmission between two such Wentzell boundaries. The mathematical system is motivated by modeling biological membranes with yet unknown effective local fluxes that are coupled to bulk diffusion.

We report nonlinear edge waves in a 2D mechanical topological insulator. A bulk lattice consists of pendulums with on-site cubic nonlinearity connected by linear springs realizing quantum spin Hall effect. We show that the nonlinear interaction between two edge modes with equal group velocities (EGV) is described by a 1D two-component coupled nonlinear Schrödinger (CNLS) equation. On the interface separating two bulk lattices with opposite spin Chern numbers, we construct linear springs such that the dispersion relation exhibits EGV points with favorable CNLS coefficients. Thus, we realize nonlinear edge waves propagating along the interface, including bright-bright (BB) edge solitons for focusing CNLS coefficients, and dark-dark edge solitons, edge domain walls, and dark-bright edge solitons for defocusing CNLS coefficients. In terms of the site amplitudes, these solutions resemble bright and dark breathers. These solutions should be topologically protected when both carrier frequencies lie within a band gap, which we explicitly show by passing BB edge solitons through compact defects on the interface. We also show energy transfer in BB edge soliton collisions with potential application to collision-based computing. Generally, vector edge solitons exhibit a large parameter space for soliton collisions, which endows mechanical devices with greater potential for information processing and other functionalities.

We investigate pseudovorticity generation in Gaussian beams undergoing self-focusing under amplitude and phase noise, using the cubic-quintic nonlinear Schrödinger equation. Pseudovorticity, defined as the curl of the optical momentum flux, characterizes local rotational flow in the absence of phase singularities. Our numerical simulations show that thermal amplitude and phase noise induce a multipolar pseudovorticity pattern. Unlike the pure cubic case, where noise asymmetries are radiated away during collapse, the quintic saturation arrests collapse and traps the noise in the resulting soliton. Hence, pseudovorticity multipoles persist, oscillating at the focusing-refocusing period. These results suggest a potential pathway for controlling local optical torque through noise engineering.

Whereas conventional Gaussian focusing gradually concentrates optical energy around the focal plane, abruptly autofocusing waves maintain a low peak intensity over most of their evolution before undergoing a sudden, high-contrast intensity surge at a prescribed focus. Since their introduction in 2010, their two-dimensional spatial realizations have enabled applications ranging from particle manipulation and material processing to terahertz generation and nonlinear optics. The recent work of Cao et al. marks the transition from (2+1)-dimensional spatial autofocusing to the full space-time domain through the experimental synthesis of spherical Airy wavepackets. This advance opens new opportunities for ultrafast structured light, tightly localized energy delivery, and nonlinear photonics.

We extend the Carter-Govan multiple-scales analysis of weakly viscous, narrowband deep-water wave packets beyond Dysthe order within the potential-flow reduction of Dias, Dyachenko, and Zakharov (DDZ). We seek to determine whether this framework generates the complex multiplier $(1 + i\beta)$ used phenomenologically to modify the nonlocal Dysthe mean-flow interaction. Although order counting places a direct viscous carrier-mean interaction at sixth order, it does not exclude an indirect fifth-order contribution arising from viscosity dependent lower-order harmonics and nonlinear interactions. We therefore derive the first correction to the induced mean flow and the complete fifth-order first-harmonic solvability condition. The resulting nonlocal terms are derivative–dependent and contain no explicit viscosity, excluding the proposed indirect mechanism within the DDZ framework. At sixth order, a restricted calculation of the nonlinear viscous block isolates a direct carrier-mean contribution with the same operator structure as the imaginary component of the prescribed mean-flow correction. Independently, a finite-adjustment-time model yields an exact, frequency dependent mean-flow response. Its low-frequency expansion produces the multiplier $ 1 + i\beta_{\mathrm{eff}}(\Omega)$, with $\beta_{\mathrm{eff}}(\Omega) =\Omega\tau.$ When $\Omega\tau = \mathcal O(\epsilon)$, the resulting phase-lag correction enters at fifth order, one order beyond the leading Dysthe mean-flow interaction.

Analytical expressions for the phase variance in a nonlinear fiber optic system based on phase-shift keying are developed. The Gauss-Hermite functions are used as the orthogonal basis to represent the noise field. Number of degrees of freedom (DOF) to accurately model the phase variance is estimated. The amplifier noise excites higher order Gauss-Hermite noise modes and the nonlinear mixing of a signal pulse and higher order Gauss-Hermite noise mode leads to new noise fields which enhance the nonlinear phase noise. The higher order noise modes propagate linearly and enhance the linear phase noise if the matched filter is not used at the receiver. Analytical expression for the optimum launch power is developed taking into account the linear and nonlinear phase noise.

We present the first demonstration of transition to spatiotemporal chaos with multiple colliding pulse sequences in systems described by perturbed cubic nonlinear Schrödinger (NLS) equations. For this purpose, we consider propagation of multiple sequences of optical pulses in two distinct types of nonlinear waveguide arrays with cubic gain and loss. By employing a perturbation theory for NLS solitons, we show that the dynamics of pulse energies in the waveguide array systems is described by generalized Lotka-Volterra (LV) models, which exhibit dissipative chaos in a wide region in parameter space. We test the LV models' predictions for chaotic dynamics of pulse energies by extensive numerical simulations with perturbed systems of coupled-NLS equations. We find excellent agreement between the results of the LV and coupled-NLS models for energy dynamics in both types of waveguide array systems, despite the strong pulse pattern distortions and the strongly nonlinear nature of the dynamics.

We investigate the quantum interference and energetic phenomena associated with classical caustics in the quantum bouncing ball problem, we refer to as quantum caustics. By considering an initial Gaussian wavepacket, we show that caustics associated with the underlying classical trajectory families are exhibited. We connect the associated phase singularity chains in the vicinity of the caustic with the semiclassical Pearcey function built on the cusp catastrophe lines. We also quantify the amount of superenergy exhibited in these solutions - regions of space where the local energy exceeds the largest constituent energy eigenvalue. We give a complimentary description of the caustic and superenergy behavior using the Madelung/Bohm trajectories, which gives additional insight about the energy of the trajectories and how they traverse the phase singularity chains.