Residual symmetries and interaction solutions for the Whitham–Broer–Kaup equation

Abstract

The nonlocal residual symmetry for the Whitham–Broer–Kaup (WBK) equation is derived by the truncated Painlevé analysis. The nonlocal residual symmetry is localized to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Based on the consistent tanh expansion method (CTE), some exact interaction solutions among different nonlinear excitations are explicitly presented. Some special interaction solutions are investigated in both analytical and graphical ways.

Appendix: The proof of Theorem 1

We write down the linearized form of the enlarged system (16)

$$\begin{aligned}&\sigma ^u_t+\sigma ^v_x+u\sigma ^u_x+\sigma ^uu_x+\beta \sigma ^u_{xx}=0, \end{aligned}$$

(38a)

$$\begin{aligned}&\sigma ^v_t+\sigma ^vu_x+v\sigma ^u_x+u\sigma ^v_x+\sigma ^uv_x\nonumber \\&\quad +\,\alpha \sigma ^u_{xxx}-\beta \sigma ^v_{xx}=0, \end{aligned}$$

(38b)

$$\begin{aligned}&\sigma ^u=\frac{(f_{i,t}+\sqrt{\alpha +\beta ^2}f_{i,xx})\sigma ^{f_i}_x}{f_{i,x}^2}, \end{aligned}$$

(38c)

$$\begin{aligned}&\sigma ^v=\frac{\alpha +\beta ^2-\beta \sqrt{\alpha +\beta ^2}}{\sqrt{\alpha +\beta ^2}}\sigma ^u_x, \end{aligned}$$

(38d)

$$\begin{aligned}&\sigma ^{f_i}_x=\sigma ^{g_i}, \end{aligned}$$

(38e)

$$\begin{aligned}&\sigma ^{g_i}_x=\sigma ^{h_i}, i=1, \ldots , n. \end{aligned}$$

(38f)

To prove this theorem, we first consider the special case, i.e., for any fixed $k, c_k\ne 0$ while $c_j=0 (j\ne k)$ in Eq. (17). In this case, we obtain the localized symmetry for $u, v, f_k, g_k$ and $h_k$ from Eqs. (1), (5) and (11)

$$\begin{aligned}&\sigma ^u=2\sqrt{\alpha +\beta ^2}c_kg_k, \end{aligned}$$

(39a)

$$\begin{aligned}&\sigma ^v=2(\alpha +\beta ^2-\beta \sqrt{\alpha +\beta ^2})c_kh_k, \end{aligned}$$

(39b)

$$\begin{aligned}&\sigma ^{f_k}=-c_kf_k^2, \end{aligned}$$

(39c)

$$\begin{aligned}&\sigma ^{g_k}=\frac{\alpha +\beta ^2-\beta \sqrt{\alpha +\beta ^2}}{\sqrt{\alpha +\beta ^2}}\sigma ^u_x, \end{aligned}$$

(39d)

$$\begin{aligned}&\sigma ^{h_k}=-2c_k(g_k^2+f_kh_k). \end{aligned}$$

(39e)

For $j\ne k$, we eliminate u through Eq. (16c) by taking $i = k$ and $i = j$, respectively. Then we have

$$\begin{aligned} f_{j,xx}=-\frac{f_{j,t}}{\sqrt{\alpha +\beta ^2}}+\frac{f_{j,x}f_{k,t}}{\sqrt{\alpha +\beta ^2}f_{k,x}}+\frac{f_{j,x}f_{k,xx}}{f_{k,x}}. \end{aligned}$$

(40)

Substituting Eq. (39a) into Eq. (38c) with $i = j$ and vanishing $f_{j,xx}$ through Eq. (40), we have

$$\begin{aligned}&2c_k\sqrt{\alpha +\beta ^2}f_{j,x}f_{k,x}^2=-f_{k,x}\sigma ^{f_j}_t+f_{k,t}\sigma ^{f_j}_x\nonumber \\&\quad +\sqrt{\alpha +\beta ^2}\sigma ^{f_j}_xf_{k,xx}-\sqrt{\alpha +\beta ^2}f_{k,x}\sigma ^{f_j}_{xx}. \end{aligned}$$

(41)

It can be easily verified that equation (41) has the solution

$$\begin{aligned} \sigma ^f_j=-c_kf_jf_k. \end{aligned}$$

(42)

The symmetry for $g_j$ and $h_j$ can be easily obtained from Eqs. (38e) and (38f) with $i = j$

$$\begin{aligned}&\sigma ^g_j=-c_k(f_kg_j+f_jg_k),\nonumber \\&\quad \sigma ^h_j=-c_k(f_kh_j+g_jg_k+f_jh_k). \end{aligned}$$

(43)

After taking the linear combination of the above results for all $k = 1, 2, \ldots , n$, Theorem 1 is proved.