Papers

The formation of a superlattice pattern in two-frequency-driven Faraday waves discovered and named SSS-I by Arbell & Fineberg (1998, 2002) is investigated by means of Direct Numerical Simulations (DNS) of the full three-dimensional Navier–Stokes equations with a free surface. Two simulations with distinct quasi-hexagonal initial conditions run at a forcing amplitude $25\%$ above the Faraday-wave onset followed quite different routes, but both led eventually to the same superlattice pattern after around 250 forcing periods. This regime is inaccessible to the approximations of weak nonlinearity or viscosity. The standing-wave pattern contain rows of patches, alternating in time between hills and lakes that are connected by a long skeleton resembing the backbone of DNA strands. The patches and skeleton of the pattern can be related to its spatial Fourier decomposition, which combines hexagonal modes with a spatially and temporally subharmonic mode. One of the transition routes passes through several fairly long-lived transients including different hexagonal patterns and another superlattice pattern; the other passes only through erratic and disordered states. After another 100 periods, the pattern became unstable and was succeeded by a dynamic version of SSS-I in which the superlattice is modulated and drifts in the direction of the backbone, while preserving its basic shape. Convergence to SSS-I states both experimentally in a large geometry and numerically from two different initial conditions and in a minimal geometry demonstrates the robustness of the SSS-I pattern.

We explore theoretically the formation and evolution of spatiotemporal vortex bullets, pulses confined in both space and time while carrying orbital angular momentum, inside a graded-index multimode fiber. Through a two-fold approach, combining variational analysis with full numerical simulations, we establish the existence of a bistable vortex soliton family, characterized by distinct radial and azimuthal quantum numbers. These wave packets exhibit a Laguerre-Gaussian multi-ring topology in their spatiotemporal structure. We identify the experimental conditions necessary for the generation of such structured states of light. A stability analysis based on Vakhitov-Kolokolov criteria reveals an upper limit for the propagation constant, below which vortex bullets are found to be stable. Stationary and non-stationary dynamics of vortex bullets with different topological charges are analyzed fully, exploiting analytical techniques that are supported by full numerical simulations. These results enhance our understanding of self-trapped, spatiotemporal vortex solitons in a graded-index medium and may lead to potential applications in the area of classical and quantum information processing.

We explore the formation, stability, and propagation dynamics of light bullets carrying orbital angular momentum (vorticity) in Bessel-structured nonlinear graded-index waveguides. While these waves are prone to the collapse in uniform three-dimensional media, we demonstrate that their stabilization is provided by a squared-Bessel index profile in the plane orthogonal to the propagation direction, which may play the role of the waveguiding or antiwaveguiding potential. Different solution families are obtained, revealing a variety of power-vs.-propagation-constant diagrams, characterizing the coexistence of multiple families. We analyze the propagation dynamics in a wide experimentally accessible parameter range. Our results contribute to the understanding of nonlinear light localization in multidimensional settings and suggest new possibilities for using structured light in advanced photonic applications.

I explore Conway's Game of Life (GoL) on six composite Archimedean lattices. On the Kagome lattice, on which small gliders and puffers appear particularly frequently across inputs, I use the output of a symmetry-constrained evolutionary search algorithm to construct a novel glider gun. The glider gun comprises four interacting bouncers and stably emits a small glider every 276th generation. Serving as an extension of classical GoL, I also propose cells with a phase degree of freedom and an associated local phase rule, which on the Kagome lattice is demonstrated to host phase-periodic gliders. This enables the possibility of phase-sensitive and interference-based computations.

We study the formation and interaction of dispersive shock waves (DSWs) in the two-dimensional Toda lattice subject to wedge-type initial conditions, and show that their interaction gives rise to a discrete analog of Mach reflection for dispersive shock waves in discrete systems. The initial jump across each leg of the wedge acts locally as a Riemann problem for the one-dimensional Toda lattice, producing two oblique DSWs whose leading-edge soliton amplitude is determined explicitly by the one-dimensional Whitham modulation theory. The two-dimensional nature of the problem manifests when these oblique DSWs meet along the symmetry axis. We show that, for compressive wedges (i.e., when the initial conditions are such that two oblique DSWs that are generated propagate toward each other), a critical slope $q_{\mathrm{cr}}$ separates two regimes: in the subcritical regime ($q<q_{\mathrm{cr}}$) the interaction is resonant and it produces an expanding DSW whose amplitude, length and velocity are explicitly computed by using exact soliton solutions of the two-dimensional Toda lattice; in the supercritical regime ($q>q_{\mathrm{cr}}$) the interaction is ordinary and produces a localized peak whose amplitude is determined analytically. We also show qualitatively that a similar dichotomy between two regimes exists for expansive wedges (i.e., when the initial conditions are such that the two oblique DSWs propagate away from each other). We confirm all analytical predictions by comparing them with the results of direct numerical simulations. Finally, we show that the continuum limit of the result is consistent with the analogous theory for the Kadomtsev-Petviashvili equation, providing an independent validation of the analytical framework.

A driven optical cavity containing a nonlinear medium can support stable, soliton-like objects, bright non-diffracting spots of light on a uniform background. Importantly, the nonlinear medium need not support ordinary (propagating) spatial solitons, and cavity solitons have been both predicted and observed in a saturable absorber, and predicted to even exist in self-defocusing media. The history, present status, and physical interpretation of such cavity solitons is reviewed. Recent research predicts cavity solitons in semiconductor cavity models with properties interesting for applications to optical information processing. Cavity solitons are natural `bits' for spatial information, possessing both robustness and plasticity, and some potential applications of their unique properties are mentioned.

Two-band superconductors maintain a relative interband phase which can carry winding soliton excitations in a superconducting ring, supported by independent winding numbers in the two bands. In rings of superconductors obeying the inversion symmetry the interband phase solitons are metastable states, separated from the uniform ground state by the energy of screening currents. In this work we find that, breaking the inversion symmetry strongly enough, one can make the soliton a true ground state. In that case, a magneto-electric coupling, absent in centrosymmetric materials, contributes critically above a certain threshold, a relevant, free-energy term, odd with respect to the winding number, which biases the energy balance in favor of a particular winding sign. Once the bias outweighs the energy cost that originally made the soliton metastable, a phase soliton with a finite winding number becomes the ground state, with chirality set by the applied field. In current–flux measurements performed in equilibrium states, the effect is demonstrated by field-odd soliton branches, that replace the metastable ones existing in mesoscopic rings, built by two-component superconductors, realizing a magneto-electric analog of the Little–Parks fluxoid-branch physics in the interband relative-phase sector.

In swollen elastomers, elasticity can arrest macroscopic demixing between the polymer network and the solvent, producing stable domains of finite size. While previous theories have focused on isotropically swollen elastomers, anisotropy strongly influences the phase behavior of many related systems, such as gels and crosslinked polymer blends. Building on the approach developed in Part I of this two-part paper, we investigate the effects of two experimentally induced sources of anisotropy: uniaxial swelling and stiffness gradients. We show that uniaxial swelling can lead to the formation of stable lamellar microphases and modify phase behavior, while stiffness gradients can cause spatial variations in the characteristic microphase size.

Large natural earthquakes are typically mediated by frictional pulse-like rupture, which features a finite slipping zone. Recently, a comprehensive two-dimensional theory of frictional slip pulses has been developed. It predicts that pulses are categorically unstable rupture modes, whose evolution is intrinsically slow. Unsteady pulses satisfy an equation of motion expressed in terms of discrete observables, which is inherently related to their steady-state counterparts. The theory also predicts a transition from decaying to slowly growing pulses. Here, we perform extensive lab earthquake experiments to test the theory. The experiments confirm the theoretical predictions for pulse-like lab earthquakes over a range of prestress levels, rupture nucleation conditions and small-scale fault roughness amplitudes. Specifically, the predicted time-dependent dynamics are tested in a plane defined by the evolving pulse size and peak slip rate, along with measurements of the dimensionless growth rate of pulses, demonstrating their intrinsically slow unsteady nature. The predicted transition between decaying and growing pulses is also experimentally demonstrated. These results constitute major progress in understanding a dominant earthquake rupture mode.

We introduce a numerical direct scattering transform scheme for dark solitons of the nonlinear Schrodinger equation, enabling the identification and complete characterization of nonlinear coherent structures in defocusing media with a continuous-wave (CW) background. Our scheme is based on numerically solving the auxiliary Zakharov-Shabat scattering problem with CW boundary conditions and on analytically derived expressions that relate the elements of the transfer matrix to the scattering data for dark solitons and continuous-spectrum waves. To test our approach, we consider two analytically solvable cases of the scattering problem: i) rectangular, and ii) hyperbolic tangent hollows in the CW background, which can contain an arbitrary number of dark solitons, with known scattering data. We revisit the analytical derivations and obtain a complete set of soliton parameters represented by discrete eigenvalues and norming constants. By supplementing the direct scattering transform algorithm with high-precision arithmetic to accurately recover soliton norming constants, we provide a robust method to analyze data from numerical or natural experiments on complex wave fields in optical, hydrodynamical, and other physical systems.

We predict a new type of stable three-dimensional (3D) vortex quantum droplets in binary Bose-Einstein condensate arranged into ring clusters that per-sistently rotate in the external potential. In contrast to clusters composed from localized quantum droplets, the states introduced here represent interacting vortex lines with identical topological charges nested in common 3D envelope and existing in much broader parameter range in comparison with local-ized quantum droplets. The intricate interplay between mean-field nonlinearity of Bose-Einstein condensate, quantum fluctuations described by Lee-Huang-Yang correction, Coriolis force arising due to rotation, and external potential leads to substantial variations of cluster shape upon increase of its rotation frequency. Rotating vortex clusters bifurcate from single-vortex quantum droplets and exist only above critical value of the rotation frequency that decreases with increase of chemical potential and depends on the number of vortex lines in the cluster. The radius of the cluster decreases with increase of rotation frequency and weakly varies with chemical potential, which determines mostly localization of the individual vortices in cluster and overall width of its envelope. Vortex droplet clusters are very robust objects existing in stable form in wide intervals of the number of particles for any number (odd or even) of vortex lines forming them. Our results may open the route to observation of stable arrays of vortex lines of different configurations per-forming regular collective motion in condensate.

Spatial patterns, such as those in dryland vegetation models, have historically been studied in systems of reaction-diffusion systems with pattern onset via a Turing bifurcation from a spatially uniform state. More recently, spatial patterns have been considered in models that incorporate spatially extended interactions via nonlocal interaction kernels. It remains largely underexplored if and how the choice of nonlocal interaction kernel contributes to differences in pattern formation and persistence, particularly in models that contain competition and facilitation. Here, we investigate spatial patterns in a reaction-diffusion model for a single species that includes nonlocal competition and facilitation processes; Gaussian, exponential, algebraic, hat, and smooth hat kernels are considered as specific examples. Via a center manifold analysis, and using the relative spatial scale of competition to facilitation and the death rate as bifurcation parameters, we identify that the choice of kernel has impacts on the pattern forming bifurcation. Bifurcations using the Gaussian, exponential, and algebraic kernels largely follow expectations of Turing patterns, but patterns in the hat and smooth hat kernels can form even when the scale of competition is less than that of facilitation. The dynamics of patterns far from onset are investigated via numerical continuation methods. The model produces the so-called "Turing-before-Tipping" phenomenon demonstrating that the arrangement into spatial patterns is an effective resilience mechanism against harsh conditions. Again, there is a kernel-dependent dichotomy in pattern behavior. Early warning signs for population extinction are observed with the Gaussian, exponential, and algebraic kernels, but not under the hat or smooth hat cases.

Bright dissipative Kerr solitons generated in anomalous-dispersion microresonators underpin integrated optical frequency combs that have enabled applications including precision metrology, spectroscopy, coherent communications, and optical frequency synthesis. In contrast, Kerr resonators operating in the normal-dispersion regime can support dark solitons and platicons, but even the initiation of frequency comb generation typically requires auxiliary mechanisms, such as avoided mode crossings or pulsed pumping, to satisfy the phase-matching condition for four-wave mixing. Here we show that topological edge states in two-dimensional Kerr resonator lattices intrinsically satisfy this phase-matching condition, enabling the generation of optical frequency combs and novel coherent dissipative structures in the normal-dispersion regime without any auxiliary mechanisms. The resulting frequency combs self-organize into collective dark solitons that exhibit large-scale spatiotemporal synchronization across multiple resonators at the entire lattice edge. By tuning the pump detuning, we demonstrate the continuous evolution of topological dark solitons into topological platicons while preserving collective synchronization. Remarkably, we show that the topological lattice also supports collective bright-like soliton states despite all constituent resonators operating in the normal-dispersion regime. Our results establish topological Kerr resonator arrays as a versatile route for engineering a broad family of coherent dissipative structures, ranging from dark solitons to bright-like soliton states, within a single platform. They also provide a route toward integrated frequency-comb generation in material platforms and wavelength ranges where normal material dispersion has traditionally constrained comb generation in conventional single-resonator systems.

This work presents a new two-dimensional quantifier plane that combines normalized permutation entropy and normalized permutation mutual information, both derived from Shannon's information theory and normalized consistently through the embedding dimension of the Bandt and Pompe method. Each point on the plane simultaneously characterizes the intrinsic uncertainty of a variable and the amount of information it shares with another. We further show that a third quantity, equivalent to the conditional entropy of one variable given the other, introduces a notion of informational independence and, when both projections of a variable pair are considered jointly, reveals directionality between them. We analyze regular, chaotic, and stochastic dynamics to illustrate the relevance of the informational plane and show how these regimes evolve as a function of a coupling factor.

We analyze pattern formation from a homogeneous steady state in a one-component nonlocal adhesion model with population pressure and degenerate mobility. First, using linear stability analysis, we derive the instability threshold and a selection rule for the fastest-growing mode, and elucidate the mechanism by which the selected wavenumber shifts toward lower wavenumbers as the mean density increases. We then perform a weakly nonlinear analysis near the critical adhesion strength and derive an explicit expression for the Landau coefficient in the Stuart–Landau equation. This expression shows that the critical bifurcation is classified as supercritical or subcritical according to the mean density, the nonlinear exponent, and the second-harmonic response ratio of the kernel. Furthermore, we show that a large nonlinear exponent promotes a transition to subcriticality and confirm, through numerical bifurcation analysis and time-dependent simulations, a bifurcation structure with a fold point and the formation of mesa patterns. Finally, through the energy limit as \(m\to\infty\), we relate the observed mesa profiles to a capacity-constrained limiting structure.

We investigate the dynamics of two-component flat-top–bubble vector solitons using a variational approximation and numerical simulations. The system is described by coupled cubic-quintic nonlinear Schrödinger equations with repulsive intercomponent coupling. We introduce a self-induced mechanism in which a localized flat-top soliton (FTS) generates a density bubble in a second component: the localized density acts as a repulsive potential for the background field, creating a composite bound state. A background-subtracted formulation and normalized super-Gaussian profiles yield collective-coordinate equations for the component centers and an effective interaction potential for their relative separation. The potential forms a binding well that is nearly triangular over its global displacement range but smooth and locally parabolic near its minimum. The variational approximation consequently predicts harmonic internal oscillations whose frequency is independent of amplitude in the small-oscillation regime. Real-time simulations verify this weak-kick behavior with good agreement between the variational and numerical frequencies. For strong phase imprints, equating the exact kick energy to the variational binding depth predicts a kick scale \(K_{\mathrm{escape}}\) for the onset of partial FTS escape. Real-time simulations support this prediction: below this scale, the composite remains nearly intact, whereas above it, a substantial fraction of the localized component leaves the bubble.

We present a reduced-order semi-analytical model for reorientational nonlinearity in chiral nematic liquid crystals, showing that the chiral pitch acts as the dominant physical length scale governing the onset of nonlinear self-focusing and soliton formation. Starting from the full Frank-Oseen equation, we derive a closed-form expression for the optically induced molecular rotation that captures the essential saturable response of the medium while reducing computational cost by more than two orders of magnitude compared with standard relaxation-method solvers. Despite its simplicity, the model reproduces the essential features of the numerically obtained nonlinear refractive index, the onset of self-localization, and the transition from discrete to continuous solitons in one and two dimensions. It further predicts the formation of fully localized astigmatic nematicons with only minor shifts in the self-localization threshold due to the neglect of nonlocal effects. The proposed model provides direct physical insight into light-matter interactions with soft matter media and offers a computationally efficient tool for the design and optimization of nonlinear photonic devices.

We investigate the propagation of counter-rotating convection rolls in Rayleigh-Bénard convection initiated locally in a quiescent fluid layer under supercritical conditions. The velocity of the front separating quiescent fluid from the forming convection rolls, and the wavenumber of the convection rolls remaining behind the front, are explored. We numerically investigate fronts of forming convection rolls over five orders of magnitude of the reduced Rayleigh number, $\epsilon$, in 2D and 3D domains, for a broad range of boundary conditions, and for different front initiation approaches. In all cases, the front velocity increases as $\epsilon^{1/2}$ with increasing $\epsilon$ for $\epsilon \lesssim 1$ in agreement with predictions using the amplitude equation. The amplitude equation description of the front velocity remains accurate for $\epsilon \lesssim 10$ except when the Prandtl number is large which yields a velocity that is faster than predicted for a fluid layer far from threshold. The wavenumber of the convection rolls increases linearly with $\epsilon$ in agreement with the wavenumber that maximizes the growth rate of perturbations in the linear regime. Farther from onset, the wavenumber growth transitions to a reduced scaling of $\epsilon^{1/4}$ in agreement with predictions using the Swift-Hohenberg equation in the large $\epsilon$ limit. The scalings describing the wavenumber variation with $\epsilon$ are independent of the domain geometry, boundary conditions, and front initiation method. However, the front-selected wavenumber at criticality does not equal the critical wavenumber of the bulk instability, in general, and depends significantly upon these details. We compare our results with experimental measurements where possible.

In this paper we investigate the behavior of localized solutions, specifically solitons, in a system of three coupled waveguides. The nonlinearity is modeled by a quasi-periodic modulation influencing the interaction between the waveguides. We analyze the evolution of the soliton profiles and their dynamics under varying modulation parameters, highlighting distinct behaviors such as attraction and repulsion among solitons. Our findings reveal that the system exhibits complex behaviors, depending on the interplay between the quasi-periodic modulation and the waveguide parameters. The study contributes to understanding the impact of quasi-periodic nonlinearity on soliton dynamics in coupled waveguide systems, laying the groundwork for potential applications in nonlinear optics and photonic devices.

In this paper we study the dynamics and stability of localized solutions in a saturable nonlinear Schrödinger equation with space- and time-dependent variable coefficients. Using a variational approach and numerical simulations, we analyze the effects of different external potential configurations. Our results reveal that the stability of the solutions is highly sensitive to the modulation parameters, leading to the emergence of alternating stable and unstable regions as a function of the modulation frequency. These findings provide valuable insights into the control of localized structures in nonlinear wave systems, with potential implications for optical waveguides, Bose-Einstein condensates, and other nonlinear media.

It is known that the attractive singular inverse-square potential gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schroedinger equation. This article summarizes theoretical results which demonstrate suppression of the collapse, caused by this singular potential, and the creation of the otherwise missing ground state (GS) in a 3D gas of bosonic particles, carrying an electric dipole moment, which are pulled to the central electric charge, with repulsive contact interactions between the particles. In the mean-field approximation, the repulsive interactions are represented by the cubic term in the respective Gross-Pitaevskii (GP) equation. In addition to the GS, excited states with angular momentum are briefly considered too. Another topic considered in the article is 1D and 2D bound states in the linear Schroedinger and GP equations with the repulsive potential, which demonstrates a singularity at r –> infinity. A very recent result is that such a potential, growing faster than the negative harmonic-oscillator potential, produces a full spectrum of counter-intuitive normalizable (localized) bound states. The article puts forward perspectives for further studies of linear and nonlinear bound states existing under the action of the potentials with the singularity at r –> 0 or r –> infinity.

We explore quantum Turing pattern zoology, where the same Lindblad equation supports a morphology atlas of stripes, spots, holes, labyrinths, and defects. The stable stripe species provides a quantitatively controlled case in which morphology and Gaussian witness loss separate parametrically. In particular, visible Turing stripes can remain after two Gaussian witness margins associated with the same $k_*$ mode cross zero in a completely positive Lindblad lattice. The witness thresholds on the exact shell fall as $\mathcal{N}^{-1}$. The stripe nematic threshold tends to a nonzero value at fixed lattice size, time window, and morphology criterion. The ratio of the morphology threshold to either witness threshold therefore grows with $\mathcal{N}$. Imaging and momentum-resolved covariance measurements probe these sectors separately.

In this paper, we identify a new mechanism for rogue wave formation in a shallow-water setting. We study a bidirectional shallow-water wave field in the context of the Kaup-Boussinesq equation, and introduce a weak high-wavenumber dissipative perturbation that breaks the underlying integrability of the system. In this setting, dominant solitons grow through successive interactions with weaker, co-propagating solitons, leading to the formation of a "champion soliton" in each direction of propagation. This behaviour is in contrast to the general intuition that dissipation damps coherent structures, and instead shows that weak dissipation can induce their intensification. Moreover, we find that weak dissipation alone is not sufficient for champion soliton formation; the presence of random waves plays a crucial role in the intensification process, catalysing the transfer of energy into dominant coherent structures. While champion solitons have previously been studied in non-integrable systems, these works primarily consider perturbations introduced through modifications of the nonlinear terms (e.g., higher-order Korteweg-de Vries and Schrödinger-type models). In the present work, high-wavenumber dissipation provides a more physically natural perturbation, since such small-scale damping is a common feature in many systems.

A weakly nonlinear regime of radial tunneling is theoretically considered for a light wave in a spherical dielectric Bragg resonator containing a set of “shell” eigenmodes (both TE and TM) with different azimuthal numbers $l\geqslant 1$, which are concentrated near a defect of the Bragg structure, several layers away from the origin. Also, a single mixed radial mode with $l=1$ is located near the resonator center, either TM or TE. Nonlinearity of Kerr type in the main approximation is actual for the central mode only, while all the shell modes remain in the linear regime. The tunneling occurs between the central mode and the shell $l=1$ mode of the same symmetry. Nontrivial part of the dynamics of optical field is described by a Hamiltonian system of ordinary differential equations for complex vectors ${\bf C}_{1}(t)$ and ${\bf C}_{2}(t)$, which determine the magnitude and spatial orientation of the wave structures at the center and at the shell, respectively. Depending on “asymmetry parameter” of tunnel coupling, the system demonstrates different variants of nonlinear behavior.

We study the finite-genus algebro-geometric solutions of the Wadati-Konno-Ichikawa (WKI) equation with the saturable nonlinearity and long-time asymptotic behaviors of their short-range perturbations. First, for both the focusing and defocusing reductions, we formulate the finite-genus Baker-Akhiezer functions as explicitly solvable the matrix Riemann-Hilbert (RH) problems on the complex spectral plane and obtain theta-function representations together with the reconstruction formulae for the WKI field and the reciprocal coordinate. We then consider the Cauchy problem of the defocusing WKI equation on a finite-genus algebro-geometric background. We construct the scattering data and RH problem, and perform a Deift-Zhou nonlinear steepest descent analysis. The space-time plane is divided into two transition regions, a Zakharov-Manakov (ZM) region, and a fast-decay region. The leading term is a phase-shifted finite-genus WKI solution. The transition corrections are governed by a Painlevé-XXXIV model, while the ZM radiation is described by parabolic-cylinder functions. The reciprocal-coordinate asymptotics are obtained simultaneously.

We study integrable variable-mass sine-Gordon model (vmSG) with point defects. Using the Lax and Bäcklund-gauge formulations, we construct type-I and type-II defect matrices, derive the corresponding sewing conditions, and generate the bulk and defect contributions to an infinite hierarchy of conserved charges. The lowest members reduce to the standard sine-Gordon energy and momentum in the homogeneous limit, while for inhomogeneous backgrounds they define integrability-generated energy- and momentum-type quantities. Defect compatibility imposes matching conditions on the variable-mass functions across the defect. An analytical transmission factor $z$ characterizes soliton transmission, topological conversion, and absorption/emission processes. We then deform the type-I sewing conditions by parameters $\alpha$ and $\beta$ and derive the associated defect anomalies. Consistent one-soliton transmission is recovered for $\alpha \beta =1$, whereas for $\alpha \beta \neq 1$ parity-centered kink-kink and kink-antikink transmissions display vanishing lowest order integrated anomalies but no generic vanishing at higher order. Numerical simulations reproduce the integrable transmission/conversion regimes and show that non-integrable defects generate weak radiative tails and lowest order quasi-conserved charges, while quasi-conservation of the higher order charges remains unestablished. These results elucidate how exact defect integrability deforms into charge-dependent quasi-conservation and establish a framework for defect-controlled soliton transport in inhomogeneous media, with potential applications to nonuniform Josephson junctions, magnetic and nonlinear-optical systems, and effective molecular and DNA models.

We investigate vector Akhmediev breathers (ABs) in the degenerate and nondegenerate regimes for the coupled Sasa-Satsuma system describing two coupled ultrashort optical pulse envelopes. Based on the carrier-wavenumber relations of the two-component background, we examine three configurations: a single vanishing wavenumber, a pair of opposite wavenumbers, and two nonzero wavenumbers of unequal magnitudes. These configurations admit distinct maximal degrees of nondegeneracy. For each configuration, we derive the reduced spectral equations, construct the existence diagrams, and compare the resulting branches with the modulation-instability gain spectra. Owing to the distinct carrier-wavenumber symmetries, nondegenerate-type modes can occur in spectrally degenerate regions, whereas degenerate-type ones may also appear in spectrally nondegenerate regions. Moreover, all admissible branches in the degenerate regimes satisfy the state-transition condition and correspond to periodic-wave states, while such branches are also present in nondegenerate regimes. Direct numerical simulations of representative solutions further confirm the analytical predictions.

In this paper, we study the emergence and dynamics of pure-quartic optical dispersive shock waves governed by the nonlinear Schrödinger (NLS) equation with self-defocusing Kerr nonlinearity and fourth-order dispersion. The corresponding dispersionless hydrodynamic system reveals a distinct mechanism for wave breaking, driven by the nonlinear self-steepening of the hydrodynamic velocity. We illustrate this mechanism through optical dam-break configurations described by Riemann problems and show that, in contrast to the classical quadratic NLS equation, wave breaking occurs prior to wave splitting even in the small amplitude regime. Numerical simulations of the pure-quartic NLS (PQNLS) equation and its hydrodynamic approximation confirm these qualitatively distinct dynamical regimes.

We construct novel higher-order lump solutions of the modified Kadomtsev-Petviashvili-I (mKP-I) equation by applying the generalized long-wave limit method with spectral perturbations. By tuning the phase parameters in the soliton solution, we obtain second- and third-order lump solutions, which, to the best of our knowledge, are reported here for the first time for the mKP-I equation. A detailed asymptotic analysis reveals that these degenerate lumps exhibit anomalous scattering analogous to that in the KP-I equation. However, in contrast to the KP-I case, no complete energy equipartition occurs after the collision, and the two lumps remain distinguishable in the subleading terms of their asymptotic amplitudes. This distinction highlights a difference in the interaction dynamics between the modified and standard KP-I equations.

New types of soliton and periodic waves are identified for a nonlinear dissipative medium where the pulse propagation is governed by the cubic complex Ginzburg-Landau equation. We find that the dynamical equation for the pulse amplitude supports two distinct types of kink and antikink solitons with different functional forms. It is found that the obtained kink and antikink soliton waveforms occur under the same fixed inverse velocity. The results also indicate that the periodic waves can propagate with variety of wave forms such as sn, cn, dn and their rational forms as well. It is also shown that in the long-wave limit, the derived periodic waves degenerate into different bright and dark soliton pulses. The stability analysis based on the theory of dispersive waves in nonlinear optics is developed. It is shown that some elliptic and soliton solutions are quasi-stable in the context of passive mode locking lasers described by complex Ginzburg-Landau equation.