The Hirota’s direct method for multiple-soliton solutions for three model equations of shallow water waves

Abstract

Multiple-soliton solutions for three model equations for shallow water waves are determined. The three models are completely integrable. The Hirota bilinear method is used to determine multiple-soliton solutions of sech-squared type for these equations. The tanh–coth method is used to obtain single soliton solutions and other solutions for these three models. The three models have different linear dispersion relations, but possess the same coefficients for the polynomials of exponentials.

Introduction

Hietarinta [1], [2] established a model equation for shallow water waves. The model is proved by Hietarinta [1], [2] and by Hereman et al. [3], [4], [5], [6] to be completely integrable and it admits N-soliton solutions. In this work, we extend this model to other two completely integrable models for shallow water waves. The Hirota bilinear method [7], [8], [9], [10] is widely used to handle problems for multiple-soliton solutions. The Hirota’s bilinear method is powerful and rather heuristic and possesses significant features that make it practical for the determination of multiple-soliton solutions in a direct fashion.

The KdV equation,

$$u_t+6{\mathit{uu}}_x+u_{\mathit{xxx}}=0\text{,}$$

can be expressed in terms of the bilinear operators [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]

$$D_x(D_t+D_x^3)f·f=0\text{,}$$

with the customary definition of the Hirota’s bilinear operators [7], [8], [9], [10]

$$D_t^n{D}_x^{m}a·b={\left(\frac{\mathrm{\partial }}{\mathrm{\partial }t}-\frac{\mathrm{\partial }}{\mathrm{\partial }{t}^{\prime }}\right)}^n{\left(\frac{\mathrm{\partial }}{\mathrm{\partial }x}-\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\prime }}\right)}^{m}a(x\text{,}t)b(x^{\prime }\text{,}{t}^{\prime })|x^{\prime }=x\text{,}{t}^{\prime }=t\text{.}$$

In what follows, we list some of the D-operators properties:

$$\begin{array}{cc}\hfill & \frac{D_t^{2}f·f}{f^2}=\iint u_{\mathit{tt}}\mathrm{d}x\mathrm{d}x\text{,}\hfill \\ \hfill & \frac{D_t{D}_x^{3}f·f}{f^2}=u_{\mathit{xt}}+3u\int {\mathit{xu}}_t\mathrm{d}{x}^{\prime }\text{,}\hfill \\ \hfill & \frac{D_x^{2}f·f}{f^2}=u\text{,}\hfill \\ \hfill & \frac{D_x^{4}f·f}{f^2}=u_{2x}+3u^2\text{,}\hfill \\ \hfill & \frac{D_t{D}_{x}f·f}{f^2}=\ln (f^2{)}_{\mathit{xt}}\text{,}\hfill \\ \hfill & \frac{D_x^{6}f·f}{f^2}=u_{4x}+15{\mathit{uu}}_{2x}+15u^3\text{.}\hfill \end{array}$$

The solution of Eq. (1) is of the form,

$$u(x\text{,}t)=2\frac{{\mathrm{\partial }}^2\ln f(x\text{,}t)}{\mathrm{\partial }{x}^2}\text{,}$$

where $f(x\text{,}t)$ is given by the perturbation expansion,

$$f(x\text{,}t)=1+\sum _{n=1}^{\infty }{ϵ}^n{f}_n(x\text{,}t)\text{,}$$

where $ϵ$ is a bookkeeping non-small parameter, and $f_n(x\text{,}t)\text{,}n=1\text{,}2\text{,}\dots$ are unknown functions that will be determined by substituting the last equation into the bilinear form and solving the resulting equations by equating different powers of $ϵ$ to zero.

On the other hand, Ito [11] introduced a new type of the bilinear equation given by

$$D_t(D_t+D_x^3)f·f=0\text{,}$$

where the first operator $D_x$ in (2) is replaced by the operator $D_t$. By using

$$u(x\text{,}t)=2(\ln f(x\text{,}t){)}_{\mathit{xx}}\text{,}$$

into (7), Ito established the well-known Ito equation,

$$u_{\mathit{tt}}+u_{\mathit{xxxt}}+3(2u_x{u}_t+{\mathit{uu}}_{\mathit{xt}})+3u_{\mathit{xx}}{\int }_{-\infty }^x{u}_t\mathrm{d}{x}^{\prime }=0\text{.}$$

The Ito equation (9) can be obtained by substituting

$$\frac{D_t^{2}f·f}{f^2}=\iint u_{\mathit{tt}}\mathrm{d}x\mathrm{d}x$$

and

$$\frac{D_t{D}_x^{3}f·f}{f^2}=u_{\mathit{xt}}+3u\int u_t\mathrm{d}{x}^{\prime }$$

into (7) to obtain

$$\iint u_{\mathit{tt}}\mathrm{d}x\mathrm{d}x+u_{\mathit{xt}}+3u\int u_t\mathrm{d}{x}^{\prime }=0\text{.}$$

Consequently, the Ito equation (9) is obtained by differentiating (12) twice with respect to x.

The aim of this work is twofold. The first is to establish two model equations for shallow water waves by extending the model developed by Hietarinta [1], [2]. The second goal is to apply the tanh–coth method [16], [17], [18], [19], [20], [21], [22], [23] to determine single soliton solutions and periodic solutions for the three model equations of shallow water waves, and to apply the Hirota’s method [7], [8], [9], [10], [11], [12], [13], [14], [15] and Hereman’s method [3], [4], [5], [6] to determine multiple-soliton solutions [24], [25], [26], [27], [26], [28], [29], [30].