Houria Triki

Sub-ODE method and soliton solutions for the variable-coefficient mKdV equation

In this work, an auxiliary equation is used for an analytic study on the time-variable coefficient modified Korteweg-de Vries (mKdV) equation. Five sets of new exact soliton-like solutions are obtained. The results show that the pulse parameters are time-dependent variable coefficients. Moreover, the basic conditions for the formation of derived solutions are presented.

Solitary wave solutions for a generalized KdV-mKdV equation with variable coefficients

In this work, a generalized time-dependent variable coefficients combined KdV-mKdV (Gardner) equation arising in plasma physics and ocean dynamics is studied. By means of three amplitude ansatz that possess modified forms to those proposed by Wazwaz in 2007, we have obtained the bell type solitary waves, kink type solitary waves, and combined type solitary waves solutions for the considered model. Importantly, the results show that there exist combined solitary wave solutions in inhomogeneous KdV-typed systems, after proving their existence in the nonlinear Schrodinger systems. It should be noted that, the characteristics of the obtained solitary wave solutions have been expressed in terms of the time-dependent coefficients. Moreover, we give the formation conditions of the obtained solutions for the considered KdV-mKdV equation with variable coefficients.

New solitons and periodic wave solutions for the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation

We consider a nonlinear Schrödinger type equation in (2+1) dimensions. The proposed equation describes the nonlinear spin dynamics of (2+1)-dimensional Heisenberg ferromagnetic spin chains with bilinear and anisotropic interactions in the semiclassical limit. We first construct three families of exact periodic solutions expressed in terms of Jacobi’s elliptic functions cn, sn and dn. We second consider the limit where the elliptic modulus approaches 1 to obtain bright and dark soliton solutions. Furthermore, we find a new type of soliton-like solution, illustrating the potentially rich set of wave solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Parametric conditions for the existence and uniqueness of exact solutions are presented. The derived structures of the obtained solutions offer a rich platform to study the nonlinear spin dynamics in magnetic materials.

Combined optical solitary waves of the Fokas—Lenells equation

Abstract In this work, we investigate the Fokas–Lenells equation describing the propagation of ultrashort pulses in optical fibers when certain terms of the next asymptotic order beyond those necessary for the nonlinear Schrö dinger equation are retained. In addition to group velocity dispersion and Kerr nonlinearity, the model involves both spatio-temporal dispersion and self-steepening terms. A class of exact combined solitary wave solutions of this equation is constructed for the first time, by adopting the complex envelope function ansatz. The influences of spatio-temporal dispersion on the characteristics of combined solitary waves is also discussed.

Dark solitons for a combined potential KdV and Schwarzian KdV equations with t-dependent coefficients and forcing term

In this work we formally derive the dark soliton solutions for the combined potential KdV and Schwarzian KdV equations. The combined KdV and Schwarzian KdV equations with time-dependent coefficients and forcing term are then investigated to obtain dark soliton solutions. The solitary wave ansatz is used to carry out the analysis for both models.

OPTICAL SOLITONS IN MULTI-DIMENSIONS WITH SPATIO-TEMPORAL DISPERSION AND NON-KERR LAW NONLINEARITY

This paper studies the dynamics of optical solitons in multi-dimensions with spatio-temporal dispersion and non-Kerr law nonlinearity. The integrability aspect is the main focus of this paper. Five different forms of nonlinearity are considered — Kerr law, power law, parabolic law, dual-power law and log law nonlinearity. The traveling wave hypothesis, ansatz approach and the semi-inverse variational principle are the integration tools that are adopted to retrieve the soliton solutions to the governing equation. As a result, several constraint conditions arise out of the integration process and represent necessary conditions for the existence of solitons.

Soliton solutions for a generalized fifth-order KdV equation with t-dependent coefficients

We consider a generalized fifth-order KdV equation with time-dependent coefficients exhibiting higher-degree nonlinear terms. This nonlinear evolution equation describes the interaction between a water wave and a floating ice cover and gravity-capillary waves. By means of the subsidiary ordinary differential equation method, some new exact soliton solutions are derived. Among these solutions, we can find the well known bright and dark solitons with sech and tanh function shapes, and other soliton-like solutions. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous KdV system supporting high-order dispersive and nonlinear effects.

Resonant optical solitons with parabolic and dual-power laws by semi-inverse variational principle

Abstract This paper obtains bright optical soliton solutions from resonant nonlinear Schrödinger’s equation by the aid of semi-inverse variational principle. The two forms of nonlinear media studied are parabolic and dual-power law. The necessary constraints for the existence of these solitons are also presented.

New types of chirped soliton solutions for the Fokas–Lenells equation

Purpose The purpose of this paper is to present a reliable treatment of the Fokas–Lenells equation, an integrable generalization of the nonlinear Schrodinger equation. The authors use a special complex envelope traveling-wave solution to carry out the analysis. The study confirms the accuracy and efficiency of the used method. Design/methodology/approach The proposed technique, namely, the trial equation method, as presented in this work has been shown to be very efficient for solving nonlinear equations with spatio-temporal dispersion. Findings A class of chirped soliton-like solutions including bright, dark and kink solitons is derived. The associated chirp, including linear and nonlinear contributions, is also determined for each of these optical pulses. Parametric conditions for the existence of chirped soliton solutions are presented. Research limitations/implications The paper presents a new efficient algorithm for handling an integrable generalization of the nonlinear Schrodinger equation. Practical/implications The authors present a useful algorithm to handle nonlinear equations with spatial-temporal dispersion. The method is an effective method with promising results. Social/implications This is a newly examined model. A useful method is presented to offer a reliable treatment. Originality/value The paper presents a new efficient algorithm for handling an integrable generalization of the nonlinear Schrodinger equation.

Solitary wave solutions for a higher order nonlinear Schrödinger equation

We consider a higher order nonlinear Schrodinger equation with third- and fourth-order dispersions, cubic-quintic nonlinearities, self steepening, and self-frequency shift effects. This model governs the propagation of femtosecond light pulses in optical fibers. In this paper, we investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the considered model. The derived analytical dark and bright wave solutions are expressed in terms of the model coefficients. These exact solutions are useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a higher-order nonlinear and dispersive Schrodinger system.