Solitons

A detailed survey of the technique of perturbation theory for nearly integrable systems, based upon the inverse scattering transform, and a minute account of results obtained by means of that technique and alternative methods are given. Attention is focused on four classical nonlinear equations: the Korteweg — de Vries, nonlinear Schrodinger, sine-Gordon, and Landau-Lifshitz equations perturbed by various Hamil-tonian and/or dissipative terms; a comprehensive list of physical applications of these perturbed equations is compiled. Systems of weakly coupled equations, which become exactly integrable when decoupled, are also considered in detail. Adiabatic and radiative eft'ects in dynamics of one and several solitons (both simple and compound) are analyzed. Generalizations of the perturbation theory to quasi-one-dimensional and quantum (semiclassical) solitons, as well as to nonsoliton nonlinear wave packets, are also considered. CONTENTS

We describe the measured group-velocity dispersion characteristics of several air-silica photonic crystal fibers with anomalous group-velocity dispersion at visible and near-infrared wavelengths. The values measured over a broad spectral range are compared to those predicted for an isolated strand of silica surrounded by air. We demonstrate a strictly single-mode fiber which has zero dispersion at a wavelength of 700 nm. These fibers are significant for the generation of solitons and supercontinua using ultrashort pulse sources.

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by purely nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are surveyed too, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation (BEC). The solitons are considered in one, two, and three dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of 1D solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for the theoretical and experimental studies alike. In both the 1D and 2D cases, the mechanism which creates solitons in NLs is principally different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

We present a review of gravitating particle-like and black hole solutions with non-Abelian gauge fields. The emphasis is given to the description of the structure of the solutions and to the connection with the results of flat space soliton physics. We describe the Bartnik-McKinnon solitons and the non-Abelian black holes arising in the Einstein-Yang-Mills theory, and consider their various generalizations. These include axially symmetric and slowly rotating configurations, solutions with higher gauge groups, Λ-term, dilaton, and higher curvature corrections. The stability issue is discussed as well. We also describe the gravitating generalizations for flat space monopoles, sphalerons, and Skyrmions.

We show that solitonic solutions of the classical string action on the AdS 5 × S 5 background that carry charges (spins) of the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system. We derive equations which determine the energy of these solitons as a function of spins. In the limit of large spins J, the first subleading 1/J coefficient in the expansion of the string energy is expected to be non-renormalised to all orders in the inverse string tension expansion and thus can be directly compared to the 1-loop anomalous dimensions of the corresponding composite operators in N = 4 super YM theory. We obtain a closed system of equations that determines this subleading coefficient and, therefore, the 1-loop anomalous dimensions of the dual SYM operators. We expect that an equivalent system of equations should follow from the thermodynamic limit of the algebraic Bethe ansatz for the SO(6) spin chain derived from SYM theory. We also identify a particular string solution whose classical energy exactly reproduces the one-loop anomalous dimension of a certain set of SYM operators with two independent R charges J 1 , J 2 .

We demonstrate that rogue waves provide a powerful tool to actively control a nonlinear system with minimal effort. Specifically, optical rogue waves-rare, bright flashes of broadband light arising in subthreshold supercontinuum generation-are initiated by an exceedingly weak stimulus. Using this effect, we produce an optically switchable, ultrastable, and bright supercontinuum with greatly enhanced coherence.

The analysis performed shows that in cubic photorefractive crystals (of Bi,,SiO,, and GaAs type in particular) with drift ("quasilocal") mechanism of optical nonlinearity spatial soliton propagation can be observed for reasonable external dc electric fields lo-15 kV/cm. These photorefractive solitons which can be excited at very low cw laser power are stable in a broad region of signal(soliton)/incoherent(spatially uniform) intensity ratio, and allow easy switching from bright to dark type. Original experimental results of self-channeling of submicrowatt HeNe laser beam in a cubic Bir2Ti02,, sample under external dc field are presented.

We report on an experimental observation of bound states of solitons in a passively mode-locked fiber soliton ring laser. The observed bound solitons are stable and have discrete, fixed soliton separations that are independent of the experimental conditions.

All stationary solutions to the one-dimensional nonlinear Schrödinger equation under box or periodic boundary conditions are presented in analytic form for the case of attractive nonlinearity. A companion paper has treated the repulsive case. Our solutions take the form of bounded, quantized, stationary trains of bright solitons. Among them are two uniquely nonlinear classes of nodeless solutions, whose properties and physical meaning are discussed in detail. The full set of symmetry-breaking stationary states are described by the Cn character tables from the theory of point groups. We make experimental predictions for the Bose-Einstein condensate and show that, though these are the analog of some of the simplest problems in linear quantum mechanics, nonlinearity introduces new and surprising phenomena.

The nonlinear Klein-Gordon equation is used as a vehicle to employ the tanh method and the sine-cosine method to formally derive a number of travelling wave solutions. The study features a variety of solutions with distinct physical structures. The work shows that one method complements the other, and each method gives solutions of formal properties. The obtained solutions include compactons, solitons, solitary patterns, and periodic solutions.

Synthetic aperture radar (SAR) images from ERS-1 and ERS-2 have been used to study the characteristics of internal waves in the East China Sea. Rank-ordered packets of nonlinear internal waves in the East China Sea are often observed in the SAR images, especially in the northeast of Taiwan. In this region, the internal wave "eld is very complicated, and its generation mechanisms include the in#uence of the tide and the upwelling, which is induced by the intrusion of the Kuroshio across the continental shelf. The internal wave distributions in the East and South China Seas have been compiled based on the SAR observations from satellites. The Kortweg}deVries (KdV) type equation has been used to study the evolution of internal wave packets generated in the upwelling area. Depending on the mixed layer depth, both elevation and depression waves can be generated based on numerical simulations as observed in the SAR images. The merging of two wave packets from nonlinear wave}wave interaction in the Yellow Sea has been observed in the SAR image and is demonstrated by numerical results.

For isotropic and homogeneous nonlinear left-handed materials, for which the effective medium approximation is valid, Maxwell's equations for electric and magnetic fields lead naturally, within the slowly varying envelope approximation, to a system of coupled nonlinear Schrödinger equations. This system is equivalent to the well-known Manakov model that under certain conditions, is completely integrable, and admits bright and dark soliton solutions. It is demonstrated that left-and right-handed ͑normal͒ nonlinear media may have compound dark and bright soliton solutions, respectively. These results are also supported by numerical calculations.

In this paper an algebraic method is devised to construct new exact wave soliton solutions for two generalized nonlinear Hirota-Satsuma coupled KdV systems of partial differential equations using symbolic software like Mathematica or Maple in terms of snðxÞ À cnðxÞ Jacobic elliptic functions and sec q À tanh q q-deformed hyperbolic functions based on the idea of the homogeneous balance method.

A low-loss terahertz air-core microstructure fiber is demonstrated for terahertz waveguiding. Substantially low attenuation constant less than 0.01 cm −1 has been achieved and the guiding wavelength is found to be tunable by linear scaling the fiber size. The experimental results well agree with the simulation based on the finite-difference frequency-domain method, which interprets the guiding mechanism as the antiresonant reflecting waveguiding. The simulated modal pattern shows that most terahertz field is concentrated inside the central hollow air core and is guided without outside interference, which has high potential for guiding intense terahertz waves with minimized loss.

We construct the open string boundary states corresponding to various timedependent deformations of the D-brane and explore several ways in which they may be used to study stringy soliton collective coordinate quantum dynamics. Among other things, we find that D-strings have exact moduli corresponding to arbitrary chiral excitations of the basic soliton. These are presumably the duals of the BPS-saturated excitations of the fundamental Type IIB string. These first steps in a systematic study of the dynamics and interactions of Dirichlet-brane solitons give further evidence of the consistency of Polchinski's new approach to string soliton physics.

We investigate metal-enhanced fluorescence of peridinin-chlorophyll protein coupled to silver nanowires using optical microscopy combined with spectrally and time-resolved fluorescence techniques. In particular we study two different sample geometries: first, in which the light-harvesting complexes are deposited onto silver nanowires, and second, where solution of both nanostructures are mixed prior deposition on a substrate. The results indicate that for the peridinin-chlorophyll complexes placed in the vicinity of the silver nanowires we observe higher intensities of fluorescence emission as compared to the reference sample, where no nanowires are present. Enhancement factors estimated for the sample where the light-harvesting complexes are mixed together with the silver nanowires prior deposition on a substrate are generally larger in comparison to the other geometry of a hybrid nanostructure. While fluorescence spectra are identical both in terms of overall shape and maximum wavelength for peridinin-chlorophyll-protein complexes both isolated and coupled to metallic nanostructures, we conclude that interaction with plasmon excitations in the latter remains neutral to the functionality of the biological system. Fluorescence transients measured for the PCP complexes coupled to the silver nanowires indicate shortening of the fluorescence lifetime pointing towards modifications of radiative rate due to plasmonic interactions. Our results can be applied for developing ways to plasmonically control the light-harvesting capability of photosynthetic complexes.

Motivated by the AdS/CFT correspondence, we show that there is a remarkable agreement between static supergravity solutions and extrema of a field theory potential. For essentially any function V(α) there are boundary conditions in anti de Sitter space so that gravitational solitons exist precisely at the extrema of V and have masses given by the value of V at these extrema. Based on this, we propose new positive energy conjectures. On the field theory side, each function V can be interpreted as the effective potential for a certain operator in the dual field theory.

Recent experimental and theoretical advances in the creation and description of bright matter wave solitons are reviewed. Several aspects are taken into account, including the physics of soliton train formation as the nonlinear Fresnel diffraction, soliton-soliton interactions, and propagation in the presence of inhomogeneities. The generation of stable bright solitons by means of Feshbach resonance techniques is also discussed.

We report experimental studies of spatial optical solitons in a nonlinear glass waveguide. Spatial solitons are self-trapped beams that propagate without diffraction. We demonstrate the formation of spatial solitons, and we investigate interactions between them. At high intensities two-photon absorption is observed to lead to the breakup of solitons. A beam propagation method is used to simulate the experimental conditions numerically, and the results agree well with experiment.

We investigate the entanglement characteristics of two general bimodal Bose-Einstein condensates -a pair of tunnel-coupled Bose-Einstein condensates and the atom-molecule Bose-Einstein condensate. We argue that the entanglement is only physically meaningful if the system is viewed as a bipartite system, where the subsystems are the two modes. The indistinguishibility of the particles in the condensate means that the atomic constituents are physically inaccessible and thus the degree of entanglement between individual particles, unlike the entanglement between the modes, is not experimentally relevant so long as the particles remain in the condensed state. We calculate the entanglement between the two modes for the exact ground state of the two bimodal condensates and consider the dynamics of the entanglement in the tunnel-coupled case.

By pumping a fiber in the anomalous group-velocity dispersion regime with a color-center laser, we generate a femtosecond distributed soliton spectrum with T > 100 fsec covering a wide spectral range between 1.55 < X < 1.85 jim. We present a theory to explain the spectrum and use autocorrelations and cross correlations to verify the results experimentally. Modulation instability and the soliton self-frequency shift effect initiate a multisoliton collision process that results in narrow, high-intensity solitons. An ensemble average over these solitons, which start from noise and frequency shift by different amounts, yields the observed broad spectra. We use the fiber output in pump-probe experiments, measuring the exciton ionization time in InGaAs/InP multiple quantum wells for the first time to our knowledge.

We show that quadratic solitons are equivalent to solitons of a nonlocal Kerr medium. This provides new physical insight into the properties of quadratic solitons, often believed to be equivalent to solitons of an effective saturable Kerr medium. The nonlocal analogy also allows for novel analytical solutions and the prediction of novel bound states of quadratic solitons.

During this last decade, several remarkable phenomena inherent to the nonlinear propagation of incoherent optical waves have been reported in the literature. This article is aimed at providing a generalized wave turbulence kinetic formulation of random nonlinear waves governed by the nonlinear Schrödinger equation in the presence of a nonlocal or a noninstantaneous nonlinear response function. Depending on the amount of nonlocal (noninstantaneous) nonlinear interaction and the amount of inhomogeneous (nonstationary) statistics of the incoherent wave, different types of kinetic equations are obtained. In the spatial domain, when the incoherent wave exhibits fluctuations that are statistically homogeneous in space, the relevant kinetic equation is the wave turbulence (Hasselmann) kinetic equation. It describes, in particular, the process of optical wave thermalization to thermodynamic equilibrium, which slows down significantly as the interaction becomes highly nonlocal. When the incoherent wave is characterized by inhomogeneous statistical fluctuations, different forms of the Vlasov equation are derived, which depend on the amount of nonlocality in the system. This Vlasov approach describes, in particular, the processes of incoherent modulational instability and the formation of localized incoherent soliton structures. In the temporal domain, the noninstantaneous nonlinear response function is constrained by the causality condition. It turns out that the relevant kinetic equation has a form analogous to the weak Langmuir turbulence equation, which describes, in particular, the formation of nonlocalized spectral incoherent solitons. In the regime of a highly noninstantaneous nonlinear response and a stationary statistics of the incoherent wave, the weak Langmuir turbulence equation reduces to the Korteweg-de Vries equation. Conversely, in the regime of a highly noninstantaneous response in the presence of a nonstationary statistics, we derive a long-range Vlasov-like kinetic equation in the temporal domain, whose self-consistent potential is constrained by the causality condition. From a broader perspective, this work indicates that the wave turbulence theory may constitute the appropriate theoretical framework to formulate statistical nonlinear optics.

We report results of a systematic analysis of the stability of one-dimensional solitons in a model including the self-repulsive or attractive cubic nonlinearity and a linear potential represented by a periodically shaking lattice, which was recently implemented in experiments with Bose-Einstein condensates. In optics, the same model applies to undulated waveguiding arrays, which are also available to the experiment. In the case of the repulsive nonlinearity, stability regions are presented, in relevant parameter planes, for fundamental gap solitons and their two-peak and three-peak bound complexes, in the first and second finite band gaps. In the model with the attractive nonlinearity, stability regions are produced for fundamental solitons and their bound states populating the semi-infinite gap. In the first finite and semi-infinite gaps, unstable solitons gradually decay into radiation, while, in the second finite band gap, they are transformed into more complex states, which may represent new species of solitons. For a large amplitude of the rocking-lattice drive, the model is tantamount to that with a "flashing" lattice potential, which is controlled by periodic sequences of instantaneous kicks. Using this correspondence, we explain generic features of the stability diagrams for the solitons. We also derive a limit case of the latter system, in the form of coupled-mode equations with a "flashing" linear coupling.

We analyze the influence of self-steepening and higher-order dispersion on the process of optical wave thermalization. This study is aimed at developing a thermodynamic formulation of supercontinuum generation in photonic crystal fibers. In the highly nonlinear regime of supercontinuum generation, the optical field exhibits a turbulent dynamics that may be described by the kinetic wave theory. In this respect, the phenomenon of spectral broadening inherent to supercontinuum generation may be interpreted as a natural process of thermalization, which is characterized by an irreversible evolution of the optical field toward a thermodynamic equilibrium state. The numerical simulations show that, under certain conditions, the intensity of the field spontaneously concentrates around two specific frequencies. The theory reveals that these particular frequencies are selected in such a way that the corresponding wave packets propagate with the same group velocity, which is shown to also match the average group velocity of the optical field. This "velocity-locking" effect and the underlying process of spectral fission are interpreted in the light of a fundamental property of statistical equilibrium thermodynamics. It is shown that a velocity locking is required, in the sense that it prevents "a macroscopic internal motion in the wave system." Furthermore, the kinetic wave theory sheds light on the role of self-steepening in the incoherent regime of supercontinuum generation. A family of equilibrium spectra is derived in the presence of self-steepening. They are found in quantitative agreement with the numerical simulations of the generalized nonlinear Schrödinger equation, without adjustable parameters.

We study the motion of bright matter wave solitons in nonlinear potentials, produced by periodic or random spatial variations of the atomic scattering length. We obtain analytical results for the soliton motion, the radiation of matter wave, and the radiative soliton decay in such configurations of the Bose-Einstein condensate. The stable regimes of propagation are analyzed. The results are in remarkable agreement with the numerical simulations of the Gross-Pitaevskii equation with periodic or random spatial variations of the mean field interactions.

The lower branch of the dispersion relation of exciton polaritons in semiconductor microcavities, operating in the strong-coupling regime, contains sections of both positive and negative curvature along one spatial direction. We show that this leads to the existence of stable one-dimensional bright microcavity solitons supported by the repulsive polariton nonlinearity. To achieve localization along the second transverse direction we propose to create a special soliton waveguide by changing the cavity detuning and hence the boundary of the soliton existence in such a way that the solitons are allowed only within the stripe of the desired width.

The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in non-metallic solids and develop "experimental" techniques for studying nonlinear problems, continues to yield a wealth of results in the theory and applications of nonlinear Hamiltonian systems with many degrees of freedom. Inspired by the studies of this seminal model, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problemsincluding energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose-Einstein condensates (BECs) in optical lattices. BECS, in particular, due to their widely ranging and easily manipulated dynamical apparatuses-with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials-provide one of the most fertile grounds for the analysis of solitary waves and their interactions. In this paper, we review recent research on BECs in the presence of deep periodic potentials, which can be reduced to nonlinear chains in appropriate circumstances. These reductions, in turn, exhibit many of the remarkable nonlinear structures (including solitons, intrinsic localized modes, and vortices) that lie at the heart of the nonlinear science research seeded by the FPU paradigm. PACS numbers: 05.45.Yv, 03.75.Lm, 03.75.Nt, 05.30.Jp

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szegő polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szegő polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.

Observational studies show that voids are prominent features of the large-scale structure of the present-day Universe. Even though their emerging from the primordial density perturbations and evolutionary patterns differ from dark matter haloes, N-body simulations and theoretical models have shown that voids also merge together to form large void structures. In this study, following Sheth & van de Weygaert, we formulate an analytical approximate description of the hierarchical void evolution of growing voids by adopting the halo merging algorithm given by Lacey & Cole in the Einstein de Sitter (EdS) Universe. To do this, we take into account the general volume distribution of voids which consists of two main void processes: merging and collapsing. We show that the volume distribution function can be reduced to a simple form, by neglecting the collapsing void contribution since the collapse process is negligible for largesize voids. Therefore, the void volume fraction has a contribution only from growing voids. This algorithm becomes the analogue of the halo merging algorithm. Based on this growing void distribution, we obtain the void merging algorithm in which we define and formulate void merging and absorption rates, as well as void size and redshift survival probabilities and also failure rates in terms of the self-similar and currently favoured dark-energy-dominated cold dark matter models in the EdS Universe.

We construct Darboux transformations for the super-symmetric KP hierarchies of Manin–Radul and Jacobian types. We also consider the binary Darboux transformation for the hierarchies. The iterations of both type of Darboux transformations are briefly discussed.

We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multi-dimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasiperiodic, single waveguide, etc. We show that when the soliton is unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to a focusing instability, whereas violation of the spectral condition leads to a drift instability. We also present a quantitative approach that allows to predict the stability and instability strength.

Localized patterns and nonlinear oscillation formation on the bounded free surface of an ideal incompressible liquid are analytically investigated . Cnoidal modes, solitons and compactons, as traveling non-axially symmetric shapes are discused. A finite-difference differential generalized Korteweg-de Vries equation is shown to describe the three-dimensional motion of the fluid surface and the limit of long and shallow channels one reobtains the well known KdV equation. A tentative expansion formula for the representation of the general solution of a nonlinear equation, for given initial condition is introduced on a graphical-algebraic basis. The model is useful in multilayer fluid dynamics, cluster formation, and nuclear physics since, up to an overall scale, these systems display liquid free surface behavior.

In this paper, based on the regular Korteweg-de Vries (KdV) system, we study negative-order KdV (NKdV) equations, particularly their Hamiltonian structures, Lax pairs, conservation laws, and explicit multisoliton and multikink wave solutions thorough bilinear Bäcklund transformations. The NKdV equations studied in our paper are differential and actually derived from the first member in the negative-order KdV hierarchy. The NKdV equations are not only gauge equivalent to the Camassa-Holm equation through reciprocal transformations but also closely related to the Ermakov-Pinney systems and the Kupershmidt deformation. The bi-Hamiltonian structures and a Darboux transformation of the NKdV equations are constructed with the aid of trace identity and their Lax pairs, respectively. The single and double kink wave and bell soliton solutions are given in an explicit formula through the Darboux transformation. The one-kink wave solution is expressed in the form of tanh while the one-bell soliton is in the form of sech, and both forms are very standard. The collisions of two-kink wave and two-bell soliton solutions are analyzed in detail, and this singular interaction differs from the regular KdV equation. Multidimensional binary Bell polynomials are employed to find bilinear formulation and Bäcklund transformations, which produce N -soliton solutions. A direct and unifying scheme is proposed for explicitly building up quasiperiodic wave solutions of the NKdV equations. Furthermore, the relations between quasiperiodic wave solutions and soliton solutions are clearly described. Finally, we show the quasiperiodic wave solution convergent to the soliton solution under some limit conditions. periodic initial data, based on both the inverse spectral theory and algebrogeometric methods, was developed by Novikov, Dubrovin, Lax, Its, Matveev et al. . For more recent reviews on the KdV equation one may refer, for instance, to Refs. .

We show that the center-of-mass motion of ultracold attractively interacting bosons can become superballistic in the presence of decoherence, via single-, two- and/or three-body losses. In the limit of weak decoherence, we analytically solve the numerical model introduced in [Phys. Rev. A 91, 063616 (2015)]. The analytical predictions allow us to identify experimentally accessible parameter regimes for which we predict superballistic motion of the center of mass. Ultracold attractive Bose gases form weakly bound molecules, quantum matter-wave bright solitons. Our computer-simulations combine ideas from classical field methods ("truncated Wigner") and piecewise deterministic stochastic processes. While the truncated Wigner approach to use an average over classical paths as a substitute for a quantum superposition often is an uncontrolled approximation, here it predicts the exact root-mean-square width when modeling an expanding Gaussian wave packet. In the superballistic regime, the leading-order of the motion can thus be modeled as a quantum superposition of classical Gaussian random walks in velocity space.

Nonlinear evolution equations with cosine/sine compacton solutions are reviewed, including the Rosenau-Hyman equation and generalizations of Korteweg-de Vries, Camassa-Holm, Boussinesq, Benjamin-Bona-Mahony, Klein-Gordon and other equations. Each equation is generalized to three dimensions and the conditions for its cosine solitary waves to be either a compacton or a soliton are determined. Several equations claimed in the literature to be different among them are found to be equivalent.

We investigate one-dimensional (1D) and 2D radial domain-wall (DW) states in the system of two nonlinear-Schrödinger/Gross-Pitaevskii (NLS/GP) equations, which are coupled by the linear mixing and by the nonlinear XPM (cross-phase-modulation). The system has straightforward applications to two-component Bose-Einstein condensates, and to the bimodal light propagation in nonlinear optics. In the former case, the two components represent different hyperfine atomic states, while in the latter setting they correspond to orthogonal polarizations of light. Conditions guaranteeing the stability of flat continuous wave (CW) asymmetric bimodal states are established, followed by the study of families of the corresponding DW patterns. Approximate analytical solutions for the DWs are found near the point of the symmetry-breaking bifurcation of the CW states. An exact DW solution is produced for ratio 3 : 1 of the XPM and SPM coefficients. The DWs between flat asymmetric states, which are mirror images to each other, are completely stable, and all other species of the DWs, with zero crossings in one or two components, are fully unstable. Interactions between two DWs are considered too, and an effective potential accounting for the attraction between them is derived analytically. Direct simulations demonstrate merger and annihilation of the interacting DWs. The analysis is extended for the system including single-and double-peak external potentials. Generic solutions for trapped DWs are obtained in a numerical form, and their stability is investigated. An exact stable solution is found for the DW trapped by a single-peak potential. In the 2D geometry, stable two-component vortices are found, with topological charges s = 1, 2, 3. Radial oscillations of annular DW-shaped pulsons, with s = 0, 1, 2, are studied too. A linear relation between the period of the oscillations and the mean radius of the DW ring is derived analytically.

We consider the multicomponent Yajima-Oikawa (YO) system and show that the two-component YO system can be derived in a physical setting of a three-coupled nonlinear Schrödinger (3-CNLS) type system by the asymptotic reduction method. The derivation is further generalized to the multicomponent case. This set of equations describes the dynamics of nonlinear resonant interaction between a one-dimensional long wave and multiple short waves. The Painlevé analysis of the general multicomponent YO system shows that the underlying set of evolution equations is integrable for arbitrary nonlinearity coefficients which will result in three different sets of equations corresponding to positive, negative, and mixed nonlinearity coefficients. We obtain the general bright N-soliton solution of the multicomponent YO system in the Gram determinant form by using Hirota's bilin-earization method and explicitly analyze the one-and two-soliton solutions of the multicomponent YO system for the above mentioned three choices of nonlinearity coefficients. We also point out that the 3-CNLS system admits special asymptotic solitons of bright, dark, anti-dark, and gray types, when the long-wave–short-wave resonance takes place. The shortwave component solitons undergo two types of energy-sharing collisions. Specifically, in the two-component YO system, we demonstrate that two types of energy-sharing collisions—(i) energy switching with opposite nature for a particular soliton in two components and (ii) similar kind of energy switching for a given soliton in both components—result for two different choices of nonlinearity coefficients. The solitons appearing in the long-wave component always exhibit elastic collision whereas those of shortwave components exhibit standard elastic collisions only for a specific choice of parameters. We have also investigated the collision dynamics of asymptotic solitons in the original 3-CNLS system. For completeness, we explore the three-soliton interaction and demonstrate the pairwise nature of collisions and unravel the fascinating state restoration property.