Aly R. Seadawy

Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma

The Zakharov-Kuznetsov (ZK) equation is an isotropic nonlinear evolution equation, first derived for weakly nonlinear ion-acoustic waves in a strongly magnetized lossless plasma in two dimensions. In the present study, by applying the extended direct algebraic method, we found the electric field potential, electric field and magnetic field in the form of traveling wave solutions for the two-dimensional ZK equation. The solutions for the ZK equation are obtained precisely and the efficiency of the method can be demonstrated. The stability of these solutions and the movement role of the waves are analyzed by making graphs of the exact solutions.

Traveling wave solutions for some coupled nonlinear evolution equations

In the present paper, an extended algebraic method is used for constructing exact traveling wave solutions for some coupled nonlinear evolution equations. By implementing the direct algebraic method, new exact solutions of the coupled KdV equations, coupled system of variant Boussinesq equations, coupled Burgers equations and generalized coupled KdV equations are obtained. The present results describe the generation and evolution of such waves, their interactions, and their stability. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.

Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma

The nonlinear three-dimensional modified Zakharov-Kuznetsov (mZK) equation governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in a presence of a uniform magnetic field. By using the reductive perturbation procedure leads to a mZK equation governing the propagation of ion dynamics of nonlinear ion-acoustic waves in a plasma. The mZK equation has solutions that represent solitary traveling waves. We found the electrostatic field potential and electric field in form traveling wave solutions for three-dimensional mZK equation. The solutions for the mZK equation are obtained precisely and efficiency of the method can be demonstrated. The stability of solitary traveling wave solutions of the mZK equation to three-dimensional long-wavelength perturbations are investigated.

Fractional solitary wave solutions of the nonlinear higher-order extended KdV equation in a stratified shear flow

The problem formulations of models for internal solitary waves in a stratified shear flow with a free surface are presented. Solitary waves solutions are generated by deriving the nonlinear higher order of extended KdV equations for the free surface displacement. All coefficients of the nonlinear higher-order extended KdV equation are expressed in terms of integrals of the modal function for the linear long-wave theory. The electric field potential and the fluid pressure in the form of traveling wave solutions of the extended KdV equation are obtained. The stability of the obtained solutions and the movement role of the waves by making the graphs of the exact solutions are discussed and analyzed.

Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions

Abstract The generalized derivative higher order non-linear Schrödinger (DNLS) equation describes pluses propagation in optical fibers and can be regarded as a special case of the generalized higher order non-linear Schrödinger equation. We derive a Lagrangian and the invariant variational principle for DNLS equation. Using the amplitude ansatz method, we obtain the different cases of the exact bright, dark and bright–dark solitary wave soliton solutions of the generalized higher order DNLS equation. By implementing the modulation instability analysis and stability analysis solutions, the stability analysis of the obtained solutions and the movement role of the waves are analyzed. All solutions are analytic and stable.

Stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation

Abstract In the present study, the nonlinear Boussinesq type equation describe the bi-directional propagation of small amplitude long capillary–gravity waves on the surface of shallow water. By using the extended auxiliary equation method, we obtained some new soliton like solutions for the two-dimensional fourth-order nonlinear Boussinesq equation with constant coefficient. These solutions include symmetrical, non-symmetrical kink solutions, solitary pattern solutions, Jacobi and Weierstrass elliptic function solutions and triangular function solutions. The stability analysis for these solutions are discussed.

New wave solutions for the fractional-order biological population model, time fractional burgers, Drinfel’d–Sokolov–Wilson and system of shallow water wave equations and their applications

Abstract In this research, by applying the improved -expansion method, we have found the travelling and solitary wave solutions of the fractional-order biological population model, time fractional Burgers equation, the Drinfel’d–Sokolov–Wilson equation and the system of shallow water wave equations. The advantage of this method is providing a new and more general travelling wave solutions for many non-linear evolution equations, it supply three different kind of solutions in the form (the hyperbolic functions, the trigonometric functions and the rational functions). This method included the extended -expansion method when and the -expansion method when N takes only positive value and zero. All of these merits help us in survey of the physical meaning of each models mentioned above for investigating stability of these models. Rapprochement between our results and the previous renowned outcome presented.

Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications

Abstract The aim of this article is to construct some new traveling wave solutions and investigate localized structures for fourth-order nonlinear Ablowitz-Kaup-Newell-Segur (AKNS) water wave dynamical equation. The simple equation method (SEM) and the modified simple equation method (MSEM) are applied in this paper to construct the analytical traveling wave solutions of AKNS equation. The different waves solutions are derived by assigning special values to the parameters. The obtained results have their importance in the field of physics and other areas of applied sciences. All the solutions are also graphically represented. The constructed results are often helpful for studying several new localized structures and the waves interaction in the high-dimensional models.

Travelling wave solutions of the generalized nonlinear fifth-order KdV water wave equations and its stability

Abstract In the present study, by implementing the direct algebraic method, we present the traveling wave solutions for some different kinds of the Korteweg–de Vries (KdV) equations. The exact solutions of the Kawahara, fifth order KdV and generalized fifth order KdV equations are obtained. Solutions for the Kawahara, fifth order KdV and generalized fifth order KdV equations are obtained precisely and efficiency of the method can be demonstrated. The stability of these solutions and the movement role of the waves by making the graphs of the exact solutions are analyzed. All solutions are exact and stable, and have applications in physics.

Integrability of the coupled cubic–quintic complex Ginzburg–Landau equations and multiple-soliton solutions via mathematical methods

This paper is devoted to study the (1+1)-dimensional coupled cubic–quintic complex Ginzburg–Landau equations (cc–qcGLEs) with complex coefficients. This equation can be used to describe the nonline...