Nonlinear Schrödinger and Davey–Stewartson Equations

In the present study, the nonlinear modulation of transverse waves propagating in a cubically nonlinear dispersive elastic medium is studied using a multi-scale expansion of wave solutions. It is found that the propagation of quasi-monochromatic transverse waves is described by a pair of coupled nonlinear Schrodinger (CNLS) equations. In the process of deriving the amplitude equations, it is observed that for a specific choice of material constants and wavenumber, the coefficient of nonlinear terms becomes zero, and the CNLS equations are no longer valid for describing the behaviour of transverse waves. In order to balance the nonlinear effects with the dispersive effects, by intensifying the nonlinearity, a new perturbation expansion is used near the critical wavenumber. It is found that the long time behaviour of the transverse waves about the critical wavenumber is given by a pair of coupled quintic nonlinear Schrodinger (CQNLS) equations. In the absence of one of the transverse waves, the CQNLS equations reduce to the single quintic nonlinear Schrodinger (QNLS) equation which has already been obtained in the context of water waves. By using a modified form of the so-called tanh method, some travelling wave solutions of the CQNLS equations are presented.

We study radial solutions of a certain two-dimensional nonlinear Schrodinger (NLS) equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schrodinger equation with Lp estimates of Laguerre functions, we are able to prove an almost-sure global well-posedness result and the invariance of the Gibbs measure. We also discuss an application to the NLS equation without harmonic potential.

Progress in the development of new tools for modem apphed mathematics resulted in a better scientific under­ standing of nonlinear waves in various fields of nonlinear optics, hydrodynamics, and plasmas. The universahty of the equations describing wave processes is one of the most important components of this success. Nonlinear Schrodinger equation (NLS) and Korteweg de Vries equation are among best illustrations of successful application of apphed mathematics as a tool to analyze various nonlinear phenomena ranging from optical communications and Bose con­ densation to ocean waves. Solutions of nonlinear equations usually result in the formation of singularities , coherent structures or solitary waves. Examples of the corresponding phenomena can be observed in filamentation of laser beams in nonlinear media, wave breaking in hydrodynamics, collapse, and Langmuir waves in plasmas. NLS in dimension one (ID) is a basis for the description of modem optical fiber communication systems. A key technological tool for development of ultrafast high-bitrate optical communication lines is a dispersion management. A dispersion-managed optical fiber is described by NLS with periodic variation of dispersion along an optical line which dramatically reduces pulse broadening. Solitary wave solutions in such systems are important information carriers in optical lines. NLS in 2D describes a stationary self-focusing of light in optical media with Kerr nonlinearity, Bose-Einstein condensate in plane geometry as well as a self-focusing in plasma. Due to self-focusing the amplitude of an NLS solution tends to infinity after finite distance of propagation, representing a blow up phenomenon. The blow up is accompanied by a dramatic contraction of the spatial size of solution, which is called collapse. Near the collapse point NLS loses its applicability and there is usually a quahtative change in the underlying nonlinear phenomena, the new mechanisms become important such as dissipation, inelastic two- and three-body collisions, which can cause a loss of atoms from the Bose-Einstein condensate, breakdown of slow envelope approximation in nonlinear optical media and plasma density depletion. If optical power is very large in optics or if a number of particles in the BoseEinstein condensate significantly exceeds the critical number, then collapse turbulence occurs with random positions of collapses in space and time. A study of collapse regularization is important in describing collapse turbulence and to be able to go beyond an individual collapse event. 3D NLS is a model for non-stationary self-focusing of hght in optical media with Kerr nonlinearity, Bose-Einstein condensate in 3D geometry and laser-plasma interactions. Self-focusing results in collapse, which is of a different type as compare with 2D, and it is called weak collapse. NLS in 3D is a model for ultrashort pulse (optical bullet) that is a key for current developments in microfabrication of photonics devices by means of intense femtosecond laser pulses. Regularization of collapse for such pulses requires an extension of the model beyond NLS and a study of the full set of MaxwellSs equations. Maxwell's equations are also necessary to study ultrashort hght propagation through thin films and guided-wave metamaterials. New challenges for nonlinear optics and apphed mathematics include description of nonlinear optical phenomena in negative-refractive-index metamaterials, which are artificial nano-composites materials that show great promise for opening the new avenues in manipulating light. Light propagating through them could exhibit extraordinary properties and brings a wealth of novel phenomena to nonlinear optics. Electrodynamical properties of metamaterials result from many factors: plasmonic oscillations in metallic nanostructures, multi-scale interaction of near field electromagnetic resonances with optical far field dynamics, surface roughness of nanostructures, high values of local fields that cause nonlinear response and etc. It has recently been shown that mathematical modeling of hght interaction with thin film metamaterials requires the use of classical hydrodynamics models. Therefore, metamaterials represent a new research

Boundary Conditions for Infinite Conservation Laws

The gauge equivalence between a noncompact version of the Ishimori spin model and the Davey–Stewartson equation is established. Explicit relationships connecting the corresponding two sets of fields involved in these systems are obtained via any pair of complex functions satisfying an equation of the Schrödinger type for a free particle. Using these formulas, two examples of classes of nontrivial exact singular solutions to the Davey–Stewartson equation are given. One of them is of the closed stringlike type, while the other is doubly periodic and is expressed in terms of Riemann theta functions. The role played by the symmetry group associated with the gauge equivalent equations under consideration is also clarified.

The intent of this work to implement first-integral method to study traveling wave solutions of some space-time nonlinear fractional differential equations (FDEs) and present their graphical simulations for analyzing different wave profiles. We show how a specific properties of Gamma functions and wave transformation can be used to reduce a FDE to an ordinary one. This method works well and reveals distinct exact solutions which are classified into two different types, namely trigonometric function and hyperbolic function solutions. The results are also depicted graphically in both 3D and 2D for different values of associated parameters. The obtained results may be useful to understand ion-acoustic waves in plasma, shallow water waves in seas and the evolution of a wave packet in three dimensions with finite depth on water under weak nonlinearity by the space-time-fractional regularized long wave equation and space-time-fractional Davey–Stewartson equation, respectively.

The soliton propagation of the fractional-in-space nonlinear Schrodinger equation (NLSE) is much more complicated than that of the corresponding integer NLSE. The aim of this paper is to discover some novel fractal soliton propagation behaviors (FSPBs) of this fractional-in-space NLSE. Firstly, the exact solution is compared with the present numerical solution, and the validity and accuracy of the present numerical method are verified. Secondly, the effect of fractional derivatives on soliton propagation is explored through the present numerical simulation results. At the same time, the present method is extended to the three-dimensional fractional-order NLSE. Finally, some novel FSPBs of the fractional-in-space NLSE are given.

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrodinger (NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grunwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schrodinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schrödinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric/asymmetric soliton and soliton–rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.

Novel exact self-similar solitary waves in graded-index media with Kerr nonlinearity

New exact solutions for Davey–Stewartson system

The optical soliton solution of nonlinear complex models holds significant importance in nonlinear optics and communication systems. Considering nonlinear complex models, often described by equations like the nonlinear Schrodinger equation (NLSE), plays a crucial role in defining the balance between dispersive and nonlinear effects, enabling the formation and maintenance of solitons over long distances. This stability is crucial for signal integrity in optical communication systems. The investigation of optical soliton solutions from nonlinear complex models is sometimes complicated. With this in mind, we employed an effective method of extended Tanh function with a Riccatti differential equation to retrieve the dark, singular and periodic wave solutions for the weakly nonlocal nonlinear Schrodinger equation with parabolic law. The obtained solutions were verified by back-substitution in the original equations, with the aid of a Mathematica to affirm the robustness of the chosen approach. Respective 2D and 3D graphs for some of the obtained results was portrayed by choosing suitable values of the parameters that were involved. An analysis of instability that results in the modulation of the steady-state as a result of co-action between the nonlinear and dispersive effects was performed on the proposed model where the condition for stable wave under small perturbation was obtained and presented. The gain spectrum plot for the modulation instability was portrayed.

In this study, we investigate the stochastic Davey-Stewartson equation in the presence of noise. These two-dimensional integrable equations are higher-dimensional versions of the nonlinear Schrödinger equation. Davey-Stewartson equations are important in plasma physics, nonlinear optics, hydrodynamics, and other disciplines because the solutions they provide are valuable in understanding many complex physical phenomena. We employ a modified version of the G′G2 approach to handle variable-coefficient systems with imaginary components, such as nonlinear Schrödinger systems. We have discovered a wide range of precise traveling wave solutions, including solitons, kink, periodic, and rational solutions. These solutions may have a significant impact in the domains of engineering and plasma physics. The presented techniques effectively achieve a variety of exponential solutions, including bright, dark, single, rational, and periodic solitary wave solutions. We used MATLAB to simulate our findings and provide 3D, 2D, and counter graphs that illustrate the impact of noise on the precise solutions of the stochastic Davey-Stewartson problem. We apply the Galilean transformation to derive the planar dynamical system, which enhances our understanding of the system’s dynamical analysis. We also conduct a sensitivity analysis to observe the systematic response to various initial conditions. This analysis focuses on the symmetrical aspects of the system and includes the illustration of phase portraits with equilibrium points. We also investigate the chaotic behavior of the planer dynamical system by introducing an additional external force. We predict the system’s behavior to increase the value of intensity and frequency. Our analysis reveals periodic, quasi-periodic, and chaotic processes.

This study explores the dynamics of highly dispersive optical solitons in nonlinear Schrödinger equations (NLSE) with non-local self-phase modulation (SPM) and polarization-mode dispersion (PMD). These nonlinear effects significantly influence soliton propagation and stability in advanced optical communication systems. Employing the Improved Modified Extended Tanh-Function Method (IMETFM), we derive exact soliton solutions, including bright, dark, singular, and combo solitons, under specific parametric conditions. The IMETFM effectively handles the complexity of the NLSE, incorporating higher-order dispersion terms (up to sixth-order) and non-local nonlinearities. Additionally, we perform a modulation instability (MI) analysis to examine the stability of steady-state solutions. This analysis uncovers the conditions under which instabilities emerge due to the interplay between dispersion and nonlinearity. The MI study offers critical insights into the growth of wave perturbations, thereby advancing the understanding of soliton stability dynamics. Graphical representations of the solutions illustrate their behavior, emphasizing the impact of non-local SPM, PMD, and MI on soliton dynamics. These findings offer valuable insights for optimizing high-capacity optical communication systems and fiber laser technologies, with broader implications for nonlinear wave phenomena in birefringent fibers and other nonlinear physical systems. This work advances the theoretical framework for soliton dynamics and lays a foundation for future experimental validations and practical applications in nonlinear optics.

For the two-dimensional nonlinear Schrodinger equation (NSE) with an external field in polar coordinates, semiclassical solutions asymptotic in small parameter ħ, ħ → 0, have been constructed. The solutions are constructed in the class of functions which are concentrated in a neighborhood of a closed plane curve and have the form of the one-soliton solution of the one-dimensional NSE in the “fast” variable.