Numerical solution of the generalized Zakharov equation by homotopy analysis method

Abstract

In this paper, an analytic technique, namely the homotopy analysis method (HAM) is applied to obtain approximations to the analytic solution of the generalized Zakharov equation. The HAM contains the auxiliary parameter $\hslash$, which provides us with a simple way to adjust and control the convergence region of the solution series.

Introduction

In the interaction of laser–plasma the system of Zakharov equation plays an important role. More recently, some authors considered the exact and explicit solutions of the system of Zakharov equations by different methods (see [1] and the references therein). First, we explain the way of obtaining this equation.

Consider a class of nonlinear partial differential equations (NPDEs) with constant coefficients

$${\mathit{iE}}_t+P\left(E_{\mathit{xx}}+A_1{E}_{\mathit{yy}}\right)+B_1|E{|}^{2}E+C_1\mathit{EF}=0\text{,}$$

$$A_2{F}_{\mathit{tt}}+\left(F_{\mathit{xx}}-B_2{F}_{\mathit{yy}}\right)+C_2{\left(|E{|}^2\right)}_{\mathit{xx}}=0\text{,}$$

where $P\text{,}{A}_i\text{,}{B}_i\text{,}{C}_i(i=1\text{,}2)$ are real constants and

$$P\ne 0\text{,}{B}_1\ne 0\text{,}{C}_1\ne 0\text{,}{C}_2\ne 0\text{.}$$

By choosing suitable values for these constants we get different kinds of equations like the Davey–Stewartson (DS) equations and nonlinear Schrödinger (NLS). If one takes

$$\left\{\begin{array}{c}F=F(x\text{,}t)\text{,}\mathrm{i}.\mathrm{e}.\text{,}{F}_y=0\text{,}\hfill \\ P=1\text{,}{A}_1=0\text{,}{B}_1=-2\lambda \text{,}{C}_1=2\text{,}\hfill \\ A_2=-1\text{,}{C}_2=-1\text{,}\hfill \end{array}\right.$$

the Eqs. (1), (2) becomes generalized Zakharov (GZ) equations [2]

$$\begin{array}{cc}\hfill & {\mathit{iE}}_t+E_{\mathit{xx}}-2\lambda |E{|}^{2}E+2\mathit{EF}=0\text{,}\hfill \\ \hfill & F_{\mathit{tt}}-F_{\mathit{xx}}+{\left(|E{|}^2\right)}_{\mathit{xx}}=0\text{,}\hfill \end{array}$$

where E is the envelope of the high-frequency electric field, and F is the plasma density measured from its equilibrium value. This system is reduced to the classical Zakharov equations of plasma physics whenever $\lambda =0$. Due to the fact that the GZE is a realistic model in plasma [3], [4], [5], it makes sense to study the solitary-wave solutions of the GZE (see [6], [7], [8], [9], for example).

In this paper, we use homotopy analysis method for solving the GZE, which was first proposed by Shi-Jun Liao [10], [11]. The HAM has been applied successfully to many nonlinear problems in engineering and science, such as applications in the generalized Hirota–Satsuma coupled KdV equation [12], in heat radiation [13], finding solitary-wave solutions for the fifth-order KdV equation [14], finding the solutions of generalized Benjamin–Bona–Mahony equation [15], finding the root of nonlinear equations [16], finding the solitary-wave solutions for the Fitzhugh–Nagumo equation [17], unsteady boundary-layer flows over a stretching flat plate [18], exponentially decaying boundary layers [19], a nonlinear model of combined convective and radiative cooling of a spherical body [20], and many other problems (see [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], for example).