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

An overview of optical solitons was presented in this project. Starting from the nonlinear effects on the refractive index and the wave equation, the Nonlinear Schrodinger Equation (NLSE) was developed. The NLSE is capable of describing both temporal and spatial solitons of both the bright and dark types. The solutions to these NLSE were also given, and their physical properties were explored. The crux of soliton theory is that the 𝛽2 linear dispersive effects (GVD or diffraction) can be balanced by intensity dependent nonlinear effects, creating a field that does not change in shape as it propagates. This places a defining condition between the pulse width and amplitude. Simulation results of soliton propagation through a dispersive fiber were presented, and it was found that the exact initial conditions of a soliton do not have to be matched, since solitons are self-adjusting. The implementation of optical solitons for fiber optic communications were studied, and the relations between bit-rate, transmission distance, and pulse widths in a typical setup were determined. There are several challenges in the implementation, the most important is power loss in the fiber, which causes an exponential broadening of pulse width during propagation. Finally, a brief description of previous experimental results on soliton transmission in fiber optic communications was presented.

are discussed in terms of how they are influenced, but not suppressed, by a metamaterial background. It is strongly discussed that metamaterials and optical rogue waves have both been making headlines in recent years and that they are, separately, large areas of research to study. A brief background of the inevitable linkage of them is considered and important new possibilities are discussed. After this background is revealed some new rogue wave configurations combining the two areas are presented alongside a discussion of the way forward for the future. 5 Nanotechnology 28 (2017) 444001 A D Boardman et al w ( ) z z t t z k V t t z z t z z

We discuss the early history of an important field of ''sturm and drang'' in modern theory of nonlinear waves. It is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability, a phenomenon resulting in a variety of nonlinear processes such as envelope solitons, envelope shocks, freak waves, etc. Examples from water wave hydrodynamics, electrodynamics, nonlinear optics, and convection theory are given.

Using a modified version of the split-step Fourier method, we analyze the effect of noise on soliton propagation inside erbium-doped fiber amplifiers. In fact, noise from forward-propagating amplified spontaneous emission, associated with a Markov immigration process, is included in the analysis of soliton amplification. Moreover, this algorithm accounts for the real spectral gain profile of the fiber amplifier. The frequency jitter, induced during soliton amplification, is compared with the Gordon-Haus effect where optical amplifiers are considered as noisy point-like devices

A new theory of functionaf expansion is presented which makes use of formal power series in several noncommutative variables and of iterated integrals. A simple closed-form expression for the solution of a nonlinear differential equation with forcing terms is derived, which enables us to give the corresponding Volterra kernels with utmost precision. The noncommutative variables give birth to a symbolic calculus which generalizes in a nonlinear setting many features of the Laplace and Fourier transforms and which is developed in order to simplify some computations, like the so-called association of variables, related to high-order transfer functions.

In the course of the past several years, a new level of understanding has been achieved about conditions for the existence, stability, and generation of spatiotemporal optical solitons, which are nondiffracting and nondispersing wavepackets propagating in nonlinear optical media. Experimentally, effectively two-dimensional (2D) spatiotemporal solitons that overcome diffraction in one transverse spatial dimension have been created in quadratic nonlinear media. With regard to the theory, fundamentally new features of light pulses that self-trap in one or two transverse spatial dimensions and do not spread out in time, when propagating in various optical media, were thoroughly investigated in models with various nonlinearities. Stable vorticity-carrying spatiotemporal solitons have been predicted too, in media with competing nonlinearities (quadratic–cubic or cubic–quintic). This article offers an up-to-date survey of experimental and theoretical results in this field. Both achievements and outstanding difficulties are reviewed, and open problems are highlighted. Also briefly described are recent predictions for stable 2D and 3D solitons in Bose–Einstein condensates supported by full or low-dimensional optical lattices.

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 propose to explore analytically one of the most promising effects (soliton) for future applications in optical communication processing and memory devices, due to the nonlinearity inside low dimensional semiconductor systems. We focus on soliton propagation in microcavity wires. We derive analytical exact solutions that describe the stationary states of the soliton's profile in the cavity. We present a scheme that gives such profiles for a given external confining potential. The obtained profiles compare favorably with existing experimental and numerical results.

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by 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 also surveyed, 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. The solitons are considered in one, two, and three dimensions. 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 one-dimensional 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 theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

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.

We consider some nonlinear phenomena in metamaterials with negative refractive index properties. Our consideration includes a survey of previously known results as well as identification of the phenomena that are important for applications of this new field. We focus on optical behavior of thin films as well as multi-wave interactions.

The creation of stable 1D and 2D localized modes in lossy nonlinear media is a fundamental problem in optics and plasmonics. This article gives a mini review of theoretical methods elaborated on for this purpose, using localized gain applied at one or several hot spots (HS). The introduction surveys a broad class of models for which this approach was developed. Other sections focus in some detail on basic 1D continuous and discrete systems, where the results can be obtained, partly or fully, in an analytical form (and verified by comparison with numerical results), which provides deeper insight into the nonlinear dynamics of optical beams in dissipative nonlinear media. Considered, in particular, are the single and double HS in the usual waveguide with the self-focusing (SF) or self-defocusing (SDF) Kerr nonlinearity, which gives rise to rather sophisticated results in spite of apparent simplicity of the model, solitons attached to a PT -symmetric dipole embedded into the SF or SDF medium, gap solitons pinned to an HS in a Bragg grating, and discrete solitons in a 1D lattice with a hot site.

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.

Through investigating history, evolution of the concept, and development in the theories of electrons, I am convinced that what was missing in our understanding of the electron is a structure, into which all attributes of the electron could be incorporated in a self-consistent way. It is hereby postulated that the topological structure of the electron is a closed two-turn Helix (a so-called Hubius Helix) that is generated by circulatory motion of a mass-less particle at the speed of light. A formulation is presented to describe an isolated electron at rest and at high speed. It is shown that the formulation is capable of incorporating most (if not all) attributes of the electron, including spin, magnetic moment, fine structure constant, anomalous magnetic moment, and charge quantization into one concrete description of the Hubius Helix. The equations for the description emerge accordingly. Implications elicited by the postulate are elaborated. Inadequacy of the formulation is discussed.

Three different methods are applied to construct new types of solutions of nonlinear evolution equations. First, the Csch method is used to carry out the solutions; then the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions. The effectiveness of these methods is demonstrated by applications to the RKL model, the generalized derivative NLS equation. The solitary wave solutions and trigonometric function solutions are obtained. The obtained solutions are very useful in the nonlinear pulse propagation through optical fibers.

We present a new discrete Adomian decomposition method to approximate the theoretical solution of discrete nonlinear Schrödinger equations. The method is examined for plane waves and for single soliton waves in case of continuous, semidiscrete and fully discrete Schrödinger equations. Several illustrative examples and Mathematica program codes are presented. (Athanassios Bratsos), [email protected] (Matthias Ehrhardt), [email protected] (Ioannis Th. Famelis).

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.

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 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.

We investigate nonlinear f(R) theories in the Kaluza-Klein models with toroidal compactification of extra dimensions. A point-like matter source has the dust-like equation of state in our three dimensions and nonzero equations of state in the extra dimensions. We obtain solutions of linearized Einstein equations with this matter source taking into account effects of nonlinearity of the model. There are two asymptotic regions where solutions satisfy the gravitational tests at the same level of accuracy as General Relativity. According to these asymptotic regions, there are two classes of solutions. We call these solutions asymptotic latent solitons. The asymptotic latent solitons from the first class generalize the known result of the linear theory. The asymptotic black strings and black branes are particular cases of these asymptotic solutions. The second class of asymptotic solitons exists only in multidimensional nonlinear models. The main feature for both of these classes of solutions is that the matter sources have tension in the extra dimensions.

This article presents a brief review of dynamical models based on systems of linearly coupled complex Ginzburg-Landau CGL equations. In the simplest case, the system features linear gain, cubic nonlinearity possibly combined with cubic loss, and group-velocity dispersion GVD in one equation, while the other equation is linear, featuring only intrinsic linear loss. The system models a dual-core fiber laser, with a parallel-coupled active core and an additional stabilizing passive lossy one. The model gives rise to exact analytical solutions for stationary solitary pulses SPs. The article presents basic results concerning stability of the SPs; interactions between pulses are also considered, as are dark solitons holes. In the case of the anomalous GVD, an unstable stationary SP may transform itself, via the Hopf bifurcation, into a stable localized breather. Various generalizations of the basic system are briefly reviewed too, including a model with quadratic second-harmonic-generating nonlinearity and a recently introduced model of a different but related type, based on linearly coupled CGL equations with cubic-quintic nonlinearity. The latter system features spontaneous symmetry breaking of stationary SPs, and also the formation of stable breathers. Complex Ginzburg-Landau (CGL) equations are ubiquitous models of pattern formation in nonlinear dissipative media, with important applications in physics and chemistry (in particular nonlinear optics and reaction-diffusion systems). Among various patterns generated by the CGL equations, especially important are localized ones, i.e., solitary pulses (SPs). The simplest cubic CGL equation with the linear gain and nonlinear loss gives rise to the well-known exact analytical solution for the SP, but it is always unstable, as the linear gain makes its background (zero solution) unstable. Therefore, an important issue is to find physically relevant modifications of the CGL equations that would be physically relevant and admit stable (and, if possible, analytically tractable) SP solutions. A known phenomenological solution to the problem is offered by the CGL equation with the cubic-quintic (CQ), rather than cubic, nonlinearity. Another option is provided by a system of two linearly coupled CGL equations , one including the linear gain, group-velocity dispersion (GVD) and cubic nonlinearity, while the other equation is linear, featuring only loss. This system is a direct physical model of a dual-core fiber laser, composed of an active core parallel-coupled to an additional (stabilizing) passive one. An important property of the system is that it admits analytical solutions for SPs, in the case in which the zero solution is stable (hence the SP has a chance to be stable too). This article aims to give a short review of models of this type and their fundamental solutions, including the identification of stability regions for the SPs in the system's parameter space, modes of destabilization of the pulses (in particular, the model with the anomalous GVD features destabilization of the SP against oscillatory perturbations, which do not destroy the pulse but rather turn it into a stable localized breather), interactions between stable SPs, solutions of the same system in the form of dark solitons, and some other properties. Also briefly considered are recently studied models of a different but related type, which are based on linearly coupled symmetric CGL equations with the CQ nonlinearity; models of this type feature such effects as spontaneous symmetry breaking between components of the SP, and formation of stable breathers in a broad range of parameters.

The exact bright one-and two-soliton solutions of a particular type of coherently coupled nonlinear Schrödinger equations, with alternate signs of nonlinearities among the two components, are obtained using the non-standard Hirota's bilinearization method. We find that in contrary to the coherently coupled nonlinear Schrödinger equations with same signs of nonlinearities the present system supports only coherently coupled solitons arising due to an interplay between dispersion and the nonlinear effects, namely, self-phase modulation, cross-phase modulation, and four-wave mixing process, thereby depend on the phases of the two co-propagating fields. The other type of soliton, namely, incoherently coupled solitons which are insensitive to the phases of the co-propagating fields and arise in a similar kind of coherently coupled nonlinear Schrödinger equations but with same signs of nonlinearities are not at all possible in the present system. The present system can support regular solution for the choice of soliton parameters for which mixed coupled nonlinear Schrödinger equations admit only singular solution. Our analysis on the collision dynamics of the bright solitons reveals the important fact that in contrary to the other types of coupled nonlinear Schrödinger systems the bright solitons of the present system can undergo only elastic collision in spite of their multicomponent nature. We also show that regular two-soliton bound states can exist even for the choice for which the same system admits singular one-soliton solution. Another important effect identified regarding the bound solitons is that the breathing effects of these bound solitons can be controlled by tuning the additional soliton parameters resulting due to the multicom-ponent nature of the system which do not have any significant effects on bright one soliton propagation and also in soliton collision dynamics. C 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4772611]

Inspired by the Penrose-Hameroff thesis, we are intuitively led to examine an intriguing correspondence of 'induction' (by fields), with the complex phenomenon of (metabolism-sustained) consciousness: Did sequences of associated induction patterns in field-susceptible biomatter have simpler beginnings?

This paper presents the design of an index guided highly birefringent photonic crystal fiber which promises to yield very large birefringence ( ) at 1550 nm and ( ) at 1064 nm as well as large effective nonlinearity( ). Optical supercontinuum generation in the proposed fiber using a 1064 nm pump source with peak power of 1 kW has been also presented. Finite difference time domain method (FDTD) has been employed to examine the optical properties such as fiber birefringence, mode field, V-parameter, walk-off and optical nonlinearity, while the Split-step Fourier method is used to solve the nonlinear Schrödinger equation felicitating the study of supercontinuum generation. Simulation results indicate that horizontal input pulse yields superior continuum in comparison to that of the vertically polarized input. However, the broadening of the continuum is about 1450 nm in case of horizontally polarized input light whereas it is approximately 2350 nm for vertically polarized.

Darboux transformations for the AKNS/ZS system are constructed in terms of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reduction of the Darboux transformation for the nonlinear Schro ̈dinger equations with standard and anomalous dispersion is presented. Two different families of new solutions for a given seed solution of the nonlinear Schro ̈dinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matrices we present topological solutions, with different asymptotic argument for the amplitude and nonzero background. For the anomalous dispersion case they represent continuous deformations of the bright n-soliton solution, which is recovered for zero background. In particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear Schro ̈dinger equation. Associated with Jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions. The second family contains not only the solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the dark soliton solutions.

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.

The principal new point is that ultra-high spin of the elementary particles makes Einstein's gravity so strong, that its influence to metric is shifted from Planck to the Compton scale! Compatibility of the Kerr-Newman (KN) gravity with quantum theory is achieved by implementation of the supersymmetric Higgs model without modification of the Einstein-Maxwell gravity. We consider the nonperturbative bag-like solution to supersymmetric generalized LG field model, which creates a flat and supersymmetric vacuum state inside the bag, forming the Compton zone for consistent work of quantum theory. The bag is deformable, and its shape is controlled by BPS bound, providing compatibility of the bag boundary with external gravitational and electromagnetic (EM) field. In particular, for the spinning KN gravity bag takes the form of oblate disk with a circular string placed on the disk border. Excitations of the KN EM field create circular traveling waves. The super-bag solution is naturally upgraded to the Wess-Zumino supersymmetric QED model, indicating a bridge from the nonperturbative super-bag to perturbative formalism of the conventional QED.

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. .

Soliton amplification in fiber amplifiers has been analyzed using either a parabolic-gain profile (as in the complex Ginzburg-Landau equation) or a Lorentzian profile (as in the set of Maxwell-Bloch equations). For multichannel soliton amplification in erbium-doped fiber amplifiers, we present a new numerical method that takes into account the actual gain profile based on the experimental wa¨elength dependence of the transition cross sections.

In this paper, the coupled Schrödinger-Boussinesq equations (SBE) will be solved by the sech, tanh, csch, and the modified simplest equation method (MSEM). We obtain exact solutions of the nonlinear for bright, dark, and singular 1-soliton solution. Kerr law nonlinearity media are studied. Results have proven that modified simple equation method does not produce the soliton solution in general case. Solutions may find practical applications and will be important for the conservation laws for dispersive optical solitons.

We consider some problems arising from singularly perturbed general differential difference equations. First we construct (in a new way) and analyze a “fitted operator finite difference method (FOFDM)” which is first order ε-uniformly convergent. With the aim of having just one function evaluation at each step, attempts have been made to derive a higher order method via Shishkin mesh to which we refer as the “fitted mesh finite difference method (FMFDM)”. This FMFDM is a direct method and ε  -uniformly convergent with the nodal error as O(n-2ln2n)O(n-2ln2n) which is an improvement over the existing direct methods (i.e., those which do not use any acceleration of convergence techniques, e.g., Richardson’s extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type that lead the error as O(n-1lnn)O(n-1lnn) where n denotes the total number of sub-intervals of [0, 1]. Comparative numerical results are presented in support of the theory.

The equations governing weakly nonlinear modulations of N -dimensional lattices are considered using a quasi-discrete multiple-scale approach. It is found that the evolution of a short wave packet for a lattice system with cubic and quartic interatomic potentials is governed by generalized Davey-Stewartson (GDS) equations, which include mean motion induced by the oscillatory wave packet through cubic interatomic interaction. The GDS equations derived here are more general than those known in the theory of water waves because of the anisotropy inherent in lattices. Generalized Kadomtsev-Petviashvili equations describing the evolution of long wavelength acoustic modes in two and three dimensional lattices are also presented. Then the modulational instability of a Ndimensional Stokes lattice wave is discussed based on the N -dimensional GDS equations obtained. Finally, the one-and two-soliton solutions of two-dimensional GDS equations are provided by means of Hirota's bilinear transformation method.

The evolution of traveling and solitary waves in Bose-Einstein condensates ͑BECs͒ with a time-dependent scattering length in an attractive/repulsive parabolic potential is studied. The homogeneous balance principle and the F-expansion technique are used to solve the one-dimensional Gross-Pitaevskii equation with timevarying coefficients. We obtained three classes of new exact traveling wave and localized solutions. Our results demonstrate that the BEC solitary wave solutions can be manipulated and controlled by the time-dependent scattering length.

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.

This paper implemented the tanh method to solve a few coupled nonlinear wave equations in (2 ? 1)dimensions. They are the Konopelchenko-Dubrovsky equation, dispersive long wave equation and the Riemann wave equation. Additionally, the traveling wave hypothesis is used to extract a few more solutons to some of these equations. Finally, the numerical simulations supplement these analytical results.