The classification and representation of single traveling wave solutions to the generalized Fornberg–Whitham equation

Abstract

By the complete discrimination system for polynomial method, we give the classification and representation of single traveling wave solutions to the generalized Fornberg–Whitham equation with $n=1$ and $n=2$.

Introduction

The generalized Fornberg–Whitham (FW) equation is as follows

$$u_t-u_{\mathit{xxt}}+u_x={\mathit{uu}}_{\mathit{xxx}}-u^n{u}_x+3u_x{u}_{\mathit{xx}}\text{,}$$

which is a classical nonlinear dispersive equation.

In the case of n=1, the Fornberg–Whitham equation

$$u_t-u_{\mathit{xxt}}+u_x={\mathit{uu}}_{\mathit{xxx}}-{\mathit{uu}}_x+3u_x{u}_{\mathit{xx}}$$

was applied to obtain the qualitative behaviour of wave breaking [1]. As a peaked limiting form of the traveling wave solution, it appears a wave of greatest height [2], $u(x\text{,}t)=A\exp (-\frac{1}{2}|x-\frac{3}{4}t|)$, where A is an arbitrary constant. Jiang and Bi [3] studied smooth and non-smooth traveling wave solutions to the FW equation by the bifurcation method of dynamical systems. Abidi and Omrani [4], [5] gave the approximate solutions to the FW equation.

In the case of n=2, He et al. [6] suggested the modified Fornberg–Whitham equation

$$u_t-u_{\mathit{xxt}}+u_x={\mathit{uu}}_{\mathit{xxx}}-u^2{u}_x+3u_x{u}_{\mathit{xx}}$$

by modifying the nonlinear term ${\mathit{uu}}_x$ to $u^2{u}_x$. Some explicit peakon and solitary wave solutions for Eq. (3) were derived, based upon the bifurcation theory and the method of phase portrait analysis.

In this paper, the classification and representation of single traveling wave solutions to the generalized FW equation with n=1 and n=2 will be given by the complete discrimination system for polynomial method proposed by Liu [7], [8], [9], [10], [11]. The method is very simple and powerful to seek for the traveling wave solutions to nonlinear differential equations. A lot of nonlinear equations were solved by liu’s method [12], [13], [14], [15], [16]. In Refs. [17], [18], [19], Ma et al. proposed some other powerful methods to solve nonlinear problems. Of course, there exist many other interesting methods to solve nonlinear differential equations, for example, Mokhtari et al. [20], [21] applied the methods of Adomian decomposition, variational iteration, direct integration, power series and the exp-function method to find the exact solutions of the Harry Dym equation and some coupled non-linear PDEs.

Firstly, taking the traveling wave transformation $u=u(\xi )$ and $\xi =\mathit{kx}+\omega t$, the Eq. (1) is reduced to the following ODE

$$\omega u^{\prime }-k^2\omega u^{‴}+{\mathit{ku}}^{\prime }=k^3{\mathit{uu}}^{‴}-{\mathit{ku}}^n{u}^{\prime }+3k^3{u}^{\prime }{u}^{″}\text{.}$$

Integrating it, we can yield the following equation

$$u^{″}+\frac{k}{\omega +\mathit{ku}}{(u^{\prime })}^2-\frac{\frac{k}{n+1}{u}^{n+1}+(\omega +k)u+c_0}{k^2(\omega +\mathit{ku})}=0\text{,}$$

where $c_0$ is an integral constant.

Secondly, the general solutions of Eq. (5) are derived [10]

$$\pm (\xi -{\xi }_0)=\int \frac{\text{d}u}{\sqrt{\exp (-2\int p_n(u)\text{d}u)[c_1-2\int q_n(u)\exp (2\int p_n(u)\text{d}u)\text{d}u]}}\text{,}$$

where $p_n(u)=\frac{k}{\omega +\mathit{ku}}\text{,}{q}_n(u)=-\frac{\frac{k}{n+1}{u}^{n+1}+(\omega +k)u+c_0}{k^2(\omega +\mathit{ku})}\text{,}{\xi }_0$ and $c_1$ are two arbitrary constants.

Then, in Sections 2 Case one :, 3 Case two :, we will use Liu’s method to gain the classification and representation of single traveling wave solutions to the Eq. (6) with $n=1$ and $n=2$.