满足变系数方程的G展开法及RLW-Burgers方程的新精确解 The G Expansion Method of Satisfying a Variable Coefficient Equation and New Exact Solutions of Rlw-Burgers Equation

doi:10.12677/AAM.2018.71010

Advances in Applied Mathematics
Vol.07 No.01(2018), Article ID:23551,11 pages
10.12677/AAM.2018.71010

The G Expansion Method of Satisfying a Variable Coefficient Equation and New Exact Solutions of Rlw-Burgers Equation

Xin Wang

●How to Cite this Article

College of Information Science and Technology, Hainan University, Haikou Hainan

Received: Dec. 21^st, 2017; accepted: Jan. 18^th, 2018; published: Jan. 25^th, 2018

ABSTRACT

Based on the basic idea of the $(G^{'} / G)$ expansion method, we construct a kind of New G method, and make the function G satisfy a class of variable coefficient Bernoulli equation. The RLW-Burgers equation is solved by this method, and several new explicit traveling wave solutions of the equation are obtained. It has been proved that this kind of satisfying variable coefficient equation G expansion method for solving nonlinear partial differential equations solutions is feasible and effective.

Keywords:RLW-Burgers Equation, Variable Coefficient Bernoulli Equation, G Expansion Method, Exact Solutions

满足变系数方程的G展开法及RLW-Burgers 方程的新精确解

王鑫

海南大学信息科学技术学院，海南海口

收稿日期：2017年12月21日；录用日期：2018年1月18日；发布日期：2018年1月25日

摘要

本文借鉴 $(G^{'} / G)$ 展开法的基本思路，构造了一类新的G展开法，并令其中的函数G满足一类变系数Bernoulli方程。用此法对RLW-Burgers方程进行了求解，得到了该方程的多个新的显式行波解。事实证明，这类满足变系数方程的G展开法对于求解非线性偏微分方程的精确解是有效可行的。

关键词 :RLW-Burgers方程，变系数Bernoulli方程，G展开法，精确解

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

求解非线性偏微分方程的精确解是目前科技发展中解决非线性问题的关键。现在已有多种可以求得非线性偏微分方程精确解的常用方法，其中 $(G^{'} / G)$ 展开法 [1] ，是一种较为简单可行的计算方法。本文就是以 $(G^{'} / G)$ 展开法为基础，构造了一类 $(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})$ 展开法，其中 $m_{i}, n_{i} (i = 1, 2)$ 为互不相等的自由参数，并令其中的函数 $G$ 满足一类变系数Bernoulli方程。通过此展开法，求得了RLW-Burgers方程的多个新的精确解。

RLW-Burgers方程

$u_{t} + u_{x} - θ u_{x x} + u u_{x} - δ u_{x x t} = 0$ (1)

其中 $θ > 0, δ > 0$ 。该方程可以用来描述河渠中水波表面的传播性态，它的行波解的存在性及解的性质已由文献 [2] 和文献 [3] 给出；振荡激波解也由文献 [4] 得到；通过未知函数的变换及待定系数法，文献 [5] 得到了该方程的显式行波解；该方程的古典对称和势对称也在文献 [6] 中得到了研究，并通过推广的双曲函数法和Lie变换群得到了该方程的精确解；文献 [7] 通过改进的G展开法得到了方程的含参数的多种形式的精确解。

2. 满足变系数方程的G展开法

将非线性偏微分方程

$P (u, u_{x}, u_{t}, u_{x x}, u_{x t}, u_{t t}, \dots) = 0$ (2)

作行波变换，令 $u (x, t) = u (ξ)$ ， $ξ = x - c t$ ，则化为常微分方程

$P (u, u^{'}, u^{″}, \dots) = 0$ ，

设方程(2)的解为

$u (ξ) = \sum_{i = 0}^{l} a_{i} {(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{i}$ (3)

其中 $a_{i} (i = 0, 1, 2, \dots)$ 为待定的常数， $m_{1}, m_{2}, n_{1}, n_{2}$ 为互不相等的自由参数，参数 $l$ 可以通过齐次平衡法确定；其中函数 $G (ξ)$ 满足一类变系数Bernoulli方程

$G^{'} (ξ) + p (ξ) G (ξ) + q (ξ) G^{2} (ξ) = 0$ (4)

这里 $p (ξ), q (ξ)$ 均为 $ξ$ 的任意函数。通过借助符号计算软件Mathematica，我们可以得到方程(4)的解：

$G (ξ) = \frac{e^{\int_{1}^{ξ} - p (τ) d τ}}{C_{1} - \int_{1}^{ξ} (- e^{\int_{1}^{ς} - p (τ) d τ} q (ς)) d ς}$ (5)

这里 $C_{1}$ 为积分常数。

将(3)式代入(2)式，并结合(4)式，合并 ${(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{i}$ 的同幂次项，并令 ${(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{i}$ 的各次幂的系数为零，从中求出 $a_{i}, p (ξ), q (ξ)$ ，再将求得的 $p (ξ), q (ξ)$ 代入(5)式，得到 $G (ξ)$ 函数，最后将 $G (ξ)$ 与 $a_{i}$ 代入(3)式即得到方程(2)的解。

3. RLW-Burgers方程新的精确解

设方程(1)有行波解 $u = u (ξ) = u (x - c t)$ ，其中 $c$ 表示波速，从而方程(1)化为

$(c + 1) u^{'} + u u^{'} - θ u^{″} - c δ u^{‴} = 0$ ，

将上式两边关于 $ξ$ 进行积分并化简，得

$(c + 1) u + \frac{1}{2} u^{2} - θ u^{'} - c δ u^{″} = M$ ， (6)

其中M为积分常数。设方程(1)的解能够表示成多项式 $u (ξ) = \sum_{i = 0}^{l} a_{i} {(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{i}$ ，且 $G = G (ξ)$ 满足一类变系数Bernoulli方程

$G^{'} (ξ) + p (ξ) G (ξ) + q (ξ) G {(ξ)}^{2} = 0$ ，

其中 $p (ξ), q (ξ)$ 为 $ξ$ 的任意函数。利用齐次平衡法，我们有 $2 l = l + 2$ ，得 $l = 2$ 。则方程(1)的解表示为

$u (ξ) = a_{2} {(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{2} + a_{1} (\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}}) + a_{0}$ ， (7)

由方程(4)和(7)式可得

$\begin{matrix} u^{'} (ξ) = - \frac{2 a_{2} (q (ξ) (- m_{2}^{2} + p (ξ) m_{2} n_{2} + n_{2}^{2} p^{'} (ξ)) + n_{2} (m_{2} - p (ξ) n_{2}) q^{'} (ξ))}{q (ξ) (m_{2} n_{1} - m_{1} n_{2})} {(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{3} \\ + \frac{1}{q (ξ) (m_{2} n_{1} - m_{1} n_{2})} (q (ξ) (a_{1} (m_{2}^{2} - p (ξ) m_{2} n_{2} - n_{2}^{2} p^{'} (ξ)) + 2 a_{2} (m_{1} (- 2 m_{2} + p (ξ) n_{2}) \\ + n_{1} (p (ξ) m_{2} + 2 n_{2} p^{'} (ξ)))) + (a_{1} n_{2} (- m_{2} + p (ξ) n_{2}) + 2 a_{2} (m_{2} n_{1} + (m_{1} - 2 p (ξ) n_{1}) n_{2})) \\ \times q^{'} (ξ)) {(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{2} + \frac{1}{q (ξ) (m_{2} n_{1} - m_{1} n_{2})} (q (ξ) (2 a_{2} (m_{1}^{2} - p (ξ) m_{1} n_{1} - n_{1}^{2} p^{'} ( ξ ) ) \end{matrix}$

$\begin{array}{l} + a_{1} (m_{1} (p (ξ) n_{2} - 2 m_{2}) + n_{1} (p (ξ) m_{2} + 2 n_{2} p^{'} (ξ)))) \\ + (2 a_{2} n_{1} (p (ξ) n_{1} - m_{1}) + a_{1} (m_{2} n_{1} + (m_{1} - 2 p (ξ) n_{1}) n_{2})) q^{'} (ξ)) (\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}}) \\ + \frac{a_{1} (q (ξ) (m_{1}^{2} - p (ξ) m_{1} n_{1} - n_{1}^{2} p^{'} (ξ)) + n_{1} (- m_{1} + p (ξ) n_{1}) q^{'} (ξ))}{q (ξ) (m_{2} n_{1} - m_{1} n_{2})} \end{array}$

$\begin{matrix} u^{″} (ξ) = \frac{6 a_{2}}{q {(ξ)}^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} {(q (ξ) (m_{2}^{2} - p (ξ) m_{2} n_{2} - n_{2}^{2} p^{'} (ξ)) + n_{2} (- m_{2} + p (ξ) n_{2}) q^{'} (ξ))}^{2} \\ \times {(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{4} + \frac{2}{q {(ξ)}^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} (n_{2} (p (ξ) n_{2} - m_{2}) (a_{1} n_{2} (p (ξ) n_{2} - m_{2}) \\ + 2 a_{2} (2 m_{2} n_{1} + (3 m_{1} - 5 p (ξ) n_{1}) n_{2})) q^{'} {(ξ)}^{2} + q {(ξ)}^{2} (a_{1} {(- m_{2}^{2} + p (ξ) m_{2} n_{2} + n_{2}^{2} p^{'} (ξ))}^{2} \\ - a_{2} (m_{1} (10 m_{2}^{3} - 15 p (ξ) m_{2}^{2} n_{2} + m_{2} n_{2}^{2} (5 p {(ξ)}^{2} - 11 p^{'} (ξ)) + n_{2}^{3} (5 p (ξ) p^{'} (ξ) - p^{″} (ξ))) + n_{1} \\ \times (5 p {(ξ)}^{2} m_{2}^{2} n_{2} - 5 p (ξ) (m_{2}^{3} - 3 m_{2} n_{2}^{2} p^{'} (ξ)) + n_{2} (- 9 m_{2}^{2} p^{'} (ξ) + 10 n_{2}^{2} p^{'} {(ξ)}^{2} + m_{2} n_{2} p^{″} (ξ))))) \end{matrix}$

$\begin{array}{l} + q (ξ) ((2 a_{1} n_{2} (m_{2} - p (ξ) n_{2}) (- m_{2}^{2} + p (ξ) m_{2} n_{2} + n_{2}^{2} p^{'} (ξ)) + a_{2} (5 (m_{2} - p (ξ) n_{2}) (m_{2}^{2} n_{1} \\ + 3 m_{2} (m_{1} - p (ξ) n_{1}) n_{2} - p (ξ) m_{1} n_{2}^{2}) + 2 n_{2}^{2} (- 7 m_{2} n_{1} + (- 3 m_{1} + 10 p (ξ) n_{1}) n_{2}) p^{'} (ξ))) q^{'} (ξ) \\ - a_{2} n_{2} (m_{2} - p (ξ) n_{2}) (m_{2} n_{1} - m_{1} n_{2}) q^{″} (ξ))) {(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{3} + \frac{1}{q {(ξ)}^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} \\ \times (2 (a_{2} (m_{2}^{2} n_{1}^{2} + 2 m_{2} n_{1} (4 m_{1} - 5 p (ξ) n_{1}) n_{2} + (3 m_{1}^{2} - 14 p (ξ) m_{1} n_{1} + 12 p {(ξ)}^{2} n_{1}^{2}) n_{2}^{2}) \\ - a_{1} n_{2} (m_{2} - p (ξ) n_{2}) (2 m_{1} n_{2} + n_{1} (m_{2} - 3 p (ξ) n_{2}))) q^{'} {(ξ)}^{2} + q {(ξ)}^{2} (2 a_{2} (m_{1}^{2} (12 m_{2}^{2} \end{array}$

$\begin{array}{l} - 12 p (ξ) m_{2} n_{2} + n_{2}^{2} (2 p {(ξ)}^{2} - 5 p^{'} (ξ))) + n_{1}^{2} (2 p {(ξ)}^{2} m_{2}^{2} + 12 p (ξ) m_{2} n_{2} p^{'} (ξ) - 3 p^{'} (ξ) m_{2}^{2} \\ - 4 n_{2}^{2} p^{'} (ξ)) + 2 m_{2} n_{2} p^{″} (ξ)) + 2 m_{1} n_{1} (4 p {(ξ)}^{2} - n_{2}^{2} p^{'} (ξ)) - n_{2} (8 m_{2} p^{'} (ξ) + n_{2} p^{″} (ξ)))) \\ - a_{1} (m_{1} (6 m_{2}^{3} - 9 p (ξ) m_{2}^{2} n_{2} + m_{2} n_{2}^{2} (3 p {(ξ)}^{2} - 7 p^{'} (ξ)) + n_{2}^{3} (3 p (ξ) p^{'} (ξ) - p^{″} (ξ)))) \\ + n_{1} (3 p {(ξ)}^{2} m_{2}^{2} n_{2} - 3 p (ξ) (m_{2}^{3} - 3 m_{2} n_{2}^{2} p^{'} (ξ)) + n_{2} (- 5 m_{2}^{2} p^{'} (ξ) + 6 n_{2}^{2} p^{'} {(ξ)}^{2} (a_{1} ((3 (m_{2} \\ - p (ξ) n_{2}) (m_{2}^{2} n_{1} + 3 m_{2} (m_{1} - p (ξ) n_{1}) n_{2} - p (ξ) m_{1} n_{2}^{2}) - 4 n_{2}^{2} (2 m_{2} n_{1} + (m_{1} - 3 p (ξ) n_{1}) n_{2}) \end{array}$

$\begin{array}{l} - n_{2} (m_{2} - p (ξ) n_{2}) (m_{2} n_{1} - m_{1} n_{2}) q^{″} (ξ)) + 2 a_{2} (- 2 (2 m_{1}^{2} n_{2} (3 m_{2} - 2 p (ξ) n_{2}) + m_{1} n_{1} (6 m_{2}^{2} \\ - 16 p (ξ) m_{2} n_{2} + n_{2}^{2} (6 p {(ξ)}^{2} - 7 p^{'} (ξ))) + n_{1}^{2} (2 p (ξ) m_{2} (- 2 m_{2} + 3 p (ξ) n_{2}) + n_{2} (- 5 m_{2} \\ + 12 p (ξ) n_{1} - m_{1} n_{2}) (m_{2} n_{1} + (m_{1} - 2 p (ξ) n_{1}) n_{2}) q^{″} (ξ)))) {(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{2} + \frac{1}{q {(ξ)}^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} \\ \times (- 2 n_{1}) (a_{1} n_{2} (- 2 m_{2} n_{1} - (m_{1} - 3 p (ξ) n_{1}) n_{2}) + 2 a_{2} n_{1} (2 m_{1} n_{2} + n_{1} (m_{2} - 3 p (ξ) n_{2}))) q^{'} {(ξ)}^{2} \\ + q {(ξ)}^{2} (- 2 a_{2} (m_{1}^{3} (6 m_{2} - 3 p (ξ) n_{2}) + m_{1}^{2} n_{1} (3 p {(ξ)}^{2} n_{2} - 9 p (ξ) m_{2} - 7 n_{2} p^{'} ( ξ ) ) \end{array}$

$\begin{array}{l} + n_{1}^{3} (3 p (ξ) m_{2} p^{'} (ξ) + 6 n_{2} p^{'} {(ξ)}^{2} + m_{2} p^{″} (ξ)) + m_{1} n_{1}^{2} (3 p {(ξ)}^{2} m_{2} - 5 m_{2} p^{'} (ξ) + 9 p (ξ) n_{2} p^{'} (ξ) \\ - n_{2} p^{″} (ξ))) + a_{1} (m_{1}^{2} (6 m_{2}^{2} - 6 p (ξ) m_{2} n_{2} + n_{2}^{2} (p {(ξ)}^{2} - 3 p^{'} (ξ))) + n_{1}^{2} (p {(ξ)}^{2} m_{2}^{2} - m_{2}^{2} p^{'} (ξ) \\ + 6 p (ξ) p^{'} (ξ) m_{2} n_{2} + 6 n_{2}^{2} p^{'} {(ξ)}^{2} + 2 m_{2} n_{2} p^{″} (ξ)) + 2 m_{1} n_{1} (2 p {(ξ)}^{2} m_{2} n_{2} - 3 p (ξ) (m_{2}^{2} - n_{2}^{2} p^{'} (ξ)) \\ - n_{2} (4 p^{'} (ξ) m_{2} + n_{2} p^{″} (ξ))))) + q (ξ) (2 a_{2} ((3 (m_{1} - p (ξ) n_{1}) (m_{2} n_{1} (3 m_{1} - p (ξ) n_{1}) + m_{1} (m_{1} \\ - 3 p (ξ) n_{1}) n_{2}) - 2 n_{1}^{2} (5 m_{1} n_{2} + m_{1} n_{2}) q^{″} (ξ)) + a_{1} (2 (m_{1}^{2} n_{2} (2 p (ξ) n_{2} - 3 m_{2}) + n_{1}^{2} (p (ξ) m_{2} (2 m_{2} \end{array}$

$\begin{array}{l} - 3 p (ξ) n_{2}) + 2 n_{2} (m_{2} - 3 p (ξ) n_{2}) p^{'} (ξ)) + m_{1} n_{1} (8 p (ξ) m_{2} n_{2} - 3 m_{2}^{2} + n_{2}^{2} (4 p^{'} (ξ) - 3 p {(ξ)}^{2}))) \\ \times q^{'} (ξ) + (m_{2} n_{1} - m_{1} n_{2}) (m_{2} n_{1} + (m_{1} - 2 p (ξ) n_{1}) n_{2}) q^{″} (ξ)))) (\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}}) \\ + \frac{1}{q {(ξ)}^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} {(2 n_{1} - p (ξ) n_{1})}^{2} (a_{2} n_{1} - a_{1} n_{2}) q^{'} {(ξ)}^{2} + q {(ξ)}^{2} (2 a_{2} (- m_{1}^{2} + p (ξ) m_{1} n_{1} \\ + {n_{1}^{2} p^{'} (ξ))}^{2} - a_{1} (m_{1}^{3} (m_{1} (2 m_{2} (2 m_{2} - p (ξ) n_{2}) + m_{1}^{2} n_{1} (- 3 p (ξ) m_{2} + p {(ξ)}^{2} n_{2} - 3 n_{2} p^{'} ( ξ ) ) \end{array}$

$\begin{array}{l} + n_{1}^{3} (p (ξ) m_{2} p^{'} (ξ) + 2 n_{2} p^{'} {(ξ)}^{2} + m_{2} p^{″} (ξ)) + m_{1} n_{1}^{2} (p {(ξ)}^{2} m_{2} - m_{2} p^{'} (ξ) + 3 p (ξ) n_{2} p^{'} (ξ) \\ - n_{2} p^{″} (ξ)))) + q (ξ) (m_{1} - p (ξ) n_{1}) (4 a_{2} n_{1} (- m_{1}^{2} + p (ξ) m_{1} n_{1} + n_{1}^{2} p^{'} (ξ)) q^{'} (ξ) \\ + a_{1} ((m_{1}^{2} n_{2} + 3 m_{1} n_{1} (m_{2} - p (ξ) n_{2}) - n_{1}^{2} (p (ξ) m_{2} + 4 p^{'} (ξ) n_{2})) q^{'} (ξ) \\ + n_{1} (- m_{2} n_{1} + m_{1} n_{2}) q^{″} (ξ)))) \end{array}$

将上面 $u$ 的各阶导数全部代入(6)式，并合并 $(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})$ 的同幂次项，比较方程两端的系数，化简得

${(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{0} :$

$\begin{array}{l} q {(ξ)}^{2} (- 2 M + 2 (1 + c) a_{0} + a_{0}^{2}) {(m_{2} n_{1} - m_{1} n_{2})}^{2} - 8 c δ q (ξ) a_{2} n_{1} (m_{1} - p (ξ) n_{1}) (- m_{1}^{2} \\ + p (ξ) m_{1} n_{1} + n_{1}^{2} p^{'} (ξ)) q^{'} (ξ) - 2 c δ q (ξ) a_{1} (m_{1} - p (ξ) n_{1}) (m_{1}^{2} n_{2} + 3 m_{1} n_{1} (m_{2} - p (ξ) n_{2}) \\ - n_{1}^{2} (p (ξ) m_{2} + 4 n_{2} p^{'} (ξ))) q^{'} (ξ) - 4 c δ n_{1} {(m_{1} - p (ξ) n_{1})}^{2} (a_{2} n_{1} - a_{1} n_{2}) q^{'} {(ξ)}^{2} \\ - 2 c δ q {(ξ)}^{2} (2 a_{2} {(- m_{1}^{2} + p (ξ) m_{1} n_{1} + n_{1}^{2} p^{'} (ξ))}^{2} - a_{1} (m_{1}^{3} (2 m_{2} - p (ξ) n_{2}) + m_{1}^{2} n_{1} \end{array}$

$\begin{array}{l} \times (- 3 p (ξ) m_{2} + p {(ξ)}^{2} n_{2} - 3 n_{2} p^{'} (ξ)) + n_{1}^{3} (p (ξ) m_{2} p^{'} (ξ) + 2 n_{2} p^{'} {(ξ)}^{2} + m_{2} p^{″} (ξ)) \\ + m_{1} n_{1}^{2} - n_{2} p^{″} (ξ)))) + 2 q (ξ) a_{1} (m_{2} n_{1} - m_{1} n_{2}) (θ q (ξ) (- m_{1}^{2} + p (ξ) m_{1} n_{1} + n_{1}^{2} p^{'} (ξ)) \\ + n_{1} (m_{1} - p (ξ) (p {(ξ)}^{2} m_{2} - m_{2} p^{'} (ξ) + 3 p (ξ) n_{2} p^{'} (ξ) n_{1}) (θ q^{'} (ξ) + c δ q^{″} (ξ))) = 0 \end{array}$

${(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{1} :$

$\begin{array}{l} 2 (2 c δ (m_{1} - p (ξ) n_{1}) (a_{1} n_{2} (- 2 m_{2} n_{1} - (m_{1} - 3 p (ξ) n_{1}) n_{2}) + 2 a_{2} n_{1} (2 m_{1} n_{2} + n_{1} \\ \times (m_{2} - 3 p (ξ) n_{2}))) q^{'} {(ξ)}^{2} + q {(ξ)}^{2} (2 a_{2} (m_{1}^{3} (6 c δ m_{2} + (θ - 3 c δ p (ξ)) n_{2}) \\ - m_{1}^{2} n_{1} ((θ + 9 c (p (ξ) (θ - 3 c δ p (ξ)) + 7 c δ p^{'} (ξ))) + m_{1} n_{1}^{2} (3 c δ p {(ξ)}^{2} m_{2} \\ - (5 c δ m_{2} + θ n_{2}) p^{'} (ξ) + p (ξ) (θ m_{2} + 9 c δ n_{2} p^{'} (ξ)) - c δ n_{2} p^{″} ( ξ ) ) \end{array}$

$\begin{array}{l} + n_{1}^{3} (6 c δ n_{2} p^{'} {(ξ)}^{2} + m_{2} ((θ + 3 c δ p (ξ)) p^{'} (ξ) + p^{″} (ξ) c δ))) + a_{1} \\ \times (m_{1}^{2} (- 6 c δ m_{2}^{2} - 2 (θ - 3 c δ p (ξ)) m_{2} n_{2} + n_{2}^{2} (1 + c + p (ξ) (θ - c δ p (ξ)) \\ + a_{0} + 3 c δ p^{'} (ξ))) + 2 m_{1} n_{1} ((θ + 3 c δ p (ξ)) m_{2}^{2} - m_{2} n_{2} (1 + c + a_{0} + 2 c δ \\ \times (p {(ξ)}^{2} - 2 p^{'} (ξ))) + n_{2}^{2} ((- 3 c δ p (ξ) + θ) p^{'} (ξ) + c δ p^{″} (ξ))) + n_{1}^{2} \end{array}$

$\begin{array}{l} \times (a_{0} + m_{2}^{2} (1 + c + c δ p^{'} (ξ)) - 2 m_{2} n_{2} ((θ + 3 c δ p (ξ)) p^{'} (ξ) + c δ p^{″} (ξ))))) \\ + q (ξ) (- 2 a_{2} (((m_{1} - p (ξ) n_{1}) (- m_{2} n_{1} (- 9 c δ m_{1} + (θ + 3 c δ p (ξ)) n_{1}) + m_{1} \\ \times (3 c δ m_{1} + (θ - 9 c δ p (ξ)) n_{1}) n_{2}) - 2 c δ n_{1}^{2} (5 m_{1} n_{2} + n_{1} (m_{2} - 6 p (ξ) n_{2})) p^{'} (ξ)) \\ \times q^{'} (ξ) + c δ n_{1} (m_{1} - p (ξ) n_{1}) (- m_{2} n_{1} + m_{1} n_{2}) q^{″} (ξ)) + a_{1} (m_{1}^{2} n_{2} \end{array}$

$\begin{array}{l} \times ((6 c δ m_{2} + (θ - 4 c δ p (ξ)) n_{2}) q^{'} (ξ) + c δ n_{2} q^{″} (ξ)) + 2 m_{1} n_{1} ((3 c δ m_{2}^{2} \\ - 8 c δ p (ξ) m_{2} n_{2} - n_{2}^{2} (p (ξ) (θ - 3 c δ p (ξ)) + 4 c δ p^{'} (ξ))) q^{'} (ξ) \\ - c δ p (ξ) q^{″} (ξ) n_{2}^{2}) + n_{1}^{2} ((m_{2} (- (θ + 4 c δ p (ξ)) m_{2} + 2 p (ξ) (θ + 3 c δ p (ξ)) n_{2}) \\ - 4 c δ n_{2} (- 3 p (ξ) n_{2} + m_{2}) p^{'} (ξ)) q^{'} (ξ) - c δ m_{2} (- p (ξ) n_{2} + m_{2}) q^{″} (ξ))))) = 0 \end{array}$

${(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{2} :$

$\begin{array}{l} (4 c δ (- a_{2} (m_{2}^{2} n_{1}^{2} + 2 m_{2} n_{1} (4 m_{1} - 5 p (ξ) n_{1}) n_{2} + (3 m_{1}^{2} - 14 p (ξ) m_{1} n_{1} + 12 p {(ξ)}^{2} n_{1}^{2}) n_{2}^{2}) \\ + a_{1} n_{2} (m_{2} - p (ξ) n_{2}) (2 m_{1} n_{2} + n_{1} (m_{2} - 3 p (ξ) n_{2}))) q^{'} {(ξ)}^{2} + q {(ξ)}^{2} (2 a_{2} (m_{1}^{2} (- c m_{2}^{2} 24 δ \\ - 4 (- 6 c δ p (ξ) + θ) m_{2} n_{2} + n_{2}^{2} (1 + c + 2 p (ξ) (θ - 2 c δ p (ξ)) + a_{0} + 10 p^{'} (ξ) c δ)) \\ + 2 m_{1} n_{1} (2 (θ + 6 c δ p (ξ)) m_{2}^{2} - m_{2} n_{2} (1 + c + a_{0} + 8 c δ (p {(ξ)}^{2} - 2 p^{'} (ξ))) + 2 n_{2}^{2} \end{array}$

$\begin{array}{l} \times ((θ - 6 c δ p (ξ)) p^{'} (ξ) + p^{″} (ξ) c δ)) + n_{1}^{2} (- 24 c δ n_{2}^{2} p^{'} {(ξ)}^{2} + m_{2}^{2} (1 + c - 2 p (ξ) \\ \times (2 c δ p (ξ) + θ) + a_{0} + 6 c δ p^{'} (ξ)) - 4 m_{2} n_{2} ((θ + 6 c δ p (ξ)) p^{'} (ξ) + c δ p^{″} (ξ)))) \\ + a_{1} (a_{1} m_{1}^{2} n_{2}^{2} + 2 m_{1} (6 c δ m_{2}^{3} + (θ - 9 c δ p (ξ)) m_{2}^{2} n_{2} - m_{2} n_{2} (a_{1} n_{1} + n_{2} (p (ξ) (θ - 3 c δ p (ξ)) \\ + 7 c δ p^{'} (ξ))) - n_{2}^{3} ((θ - 3 c δ p (ξ)) p^{'} (ξ) + c δ p^{″} (ξ))) + n_{1} (- 2 (θ + 3 c δ p (ξ)) m_{2}^{3} \end{array}$

$\begin{array}{l} + 12 c δ n_{2}^{3} p^{'} {(ξ)}^{2} + m_{2}^{2} (a_{1} n_{1} + 2 n_{2} (p (ξ) (θ + 3 c δ p (ξ)) - 5 c δ p^{'} (ξ))) + 2 m_{2} n_{2}^{2} \\ \times ((θ + 9 c δ p (ξ)) p^{'} (ξ) + c δ p^{″} (ξ))))) + 2 q (ξ) (a_{1} ((- (m_{2} - p (ξ) n_{2}) (3 c δ m_{2}^{2} n_{1} \\ + m_{2} (9 c δ m_{1} - (θ + 9 c δ p (ξ)) n_{1}) n_{2} + (θ - 3 c δ p (ξ)) m_{1} n_{2}^{2}) + 4 c δ n_{2}^{2} (2 m_{2} n_{1} \\ + (m_{1} - 3 p (ξ) n_{1}) n_{2}) p^{'} (ξ)) q^{'} (ξ) + c δ n_{2} (m_{2} - p (ξ) n_{2}) (m_{2} n_{1} - m_{1} n_{2}) q^{″} ( ξ ) ) \end{array}$

$\begin{array}{l} + 2 a_{2} (m_{1}^{2} n_{2} ((12 c δ m_{2} + (θ - 8 c δ p (ξ)) n_{2}) q^{'} (ξ) + c δ n_{2} q^{″} (ξ)) \\ + 2 m_{1} n_{1} ((6 c δ m_{2}^{2} - 16 c δ p (ξ) m_{2} n_{2} - n_{2}^{2} (p (ξ) (θ - 6 c δ p (ξ)) + 7 c δ p^{'} (ξ))) q^{'} (ξ) \\ - c δ p (ξ) n_{2}^{2} q^{″} (ξ)) + n_{1}^{2} ((m_{2} (- (θ + 8 c δ p (ξ)) m_{2} + 2 p (ξ) (θ + 6 p (ξ) c δ) n_{2}) \\ + 2 c δ n_{2} (- 5 m_{2} + 12 p (ξ) n_{2}) p^{'} (ξ)) q^{'} (ξ) - c δ m_{2} (m_{2} - 2 p (ξ) n_{2}) q^{″} (ξ)))) = 0 \end{array}$

${(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{3} :$

$\begin{array}{l} (4 c δ n_{2} (m_{2} - p (ξ) n_{2}) (a_{1} n_{2} (- m_{2} + p (ξ) n_{2}) + 2 a_{2} (2 m_{2} n_{1} + (3 m_{1} - 5 p (ξ) n_{1}) n_{2})) q^{'} {(ξ)}^{2} \\ + 2 q {(ξ)}^{2} (- 2 c δ a_{1} {(- m_{2}^{2} + p (ξ) m_{2} n_{2} + n_{2}^{2} p^{'} (ξ))}^{2} + a_{2} (a_{1} m_{1}^{2} n_{2}^{2} + 2 m_{1} (10 c δ m_{2}^{3} \\ + (- 15 c δ p (ξ) + θ) m_{2}^{2} n_{2} - m_{2} n_{2} (a_{1} n_{1} + n_{2} (p (ξ) (θ - 5 c δ p (ξ)) + 11 c δ p^{'} (ξ))) \\ - n_{2}^{3} ((θ - 5 c δ p (ξ)) p^{'} (ξ) + c δ p^{″} (ξ))) + n_{1} (- 2 (θ + 5 c δ p (ξ)) m_{2}^{3} + 20 c δ n_{2}^{3} p^{'} {(ξ)}^{2} \\ + m_{2}^{2} (a_{1} n_{1} + 2 n_{2} (p (ξ) (θ + 5 p (ξ) c δ) - 9 c δ p^{'} (ξ))) + 2 m_{2} n_{2}^{2} ((θ + 15 c δ p (ξ)) p^{'} ( ξ ) \end{array}$

$\begin{array}{l} + c δ p^{″} (ξ))))) + 4 q (ξ) (2 c δ a_{1} n_{2} (p (ξ) n_{2} - m_{2}) (- m_{2}^{2} + p (ξ) m_{2} n_{2} + n_{2}^{2} p^{'} (ξ)) q^{'} (ξ) \\ + a_{2} ((- (m_{2} - p (ξ) n_{2}) (5 c δ m_{2}^{2} n_{1} + m_{2} (15 c δ m_{1} - (θ + 15 c δ p (ξ)) n_{1}) n_{2} \\ + (- 5 p (ξ) c δ + θ) m_{1} n_{2}^{2}) + 2 c δ n_{2}^{2} (7 m_{2} n_{1} + (3 m_{1} - 10 p (ξ) n_{1}) n_{2}) p^{'} (ξ)) q^{'} (ξ) \\ + c δ n_{2} (m_{2} - n_{2} p (ξ)) (m_{2} n_{1} - m_{1} n_{2}) q^{″} (ξ)))) = 0 \end{array}$

${(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})}^{4}$ :

$\begin{array}{l} q {(ξ)}^{2} a_{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2} - 12 c δ (q (ξ) (m_{2}^{2} - p (ξ) m_{2} n_{2} - n_{2}^{2} p^{'} (ξ)) \\ + {n_{2} (- m_{2} + n_{2} p (ξ)) q^{'} (ξ))}^{2} = 0 \end{array}$ 。

用Mathematica软件对以上代数方程组进行求解，我们得到了以下五组解：

1) $a_{2} = \frac{3 {(10 c δ m_{2} + (θ - 10 c δ C_{3}) n_{2})}^{2} {(- 10 c δ m_{2} + (θ + 10 c δ C_{3}) n_{2})}^{2}}{2500 c^{3} δ^{3} {(m_{2} n_{1} - m_{1} n_{2})}^{2}}$ ，

$a_{1} = \frac{3 (10 c δ m_{1} - (θ + 10 c δ C_{3}) n_{1}) {(10 c δ m_{2} + (θ - 10 c δ C_{3}) n_{2})}^{2} (- 10 c δ m_{2} + (θ + 10 c δ C_{3}) n_{2})}{1250 c^{3} δ^{3} {(m_{2} n_{1} - m_{1} n_{2})}^{2}}$ ，

$\begin{matrix} a_{0} = \frac{1}{2500 c^{3} δ^{3} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} (3 θ^{3} n_{1} n_{2} (20 c δ m_{2} n_{1} + (- 20 c δ m_{1} + θ n_{1}) n_{2}) \\ - 300 c^{2} δ^{2} θ^{2} (m_{2}^{2} n_{1}^{2} - 2 C_{3} m_{2} n_{1}^{2} n_{2} + (m_{1}^{2} - 2 C_{3} m_{1} n_{1} + 2 C_{3}^{2} n_{1}^{2}) n_{2}^{2}) \\ + 500 c^{3} δ^{3} (m_{2} n_{1} - m_{1} n_{2}) (- n_{1} ((5 + 5 c - 12 θ C_{3}) m_{2} + 12 θ C_{3}^{2} n_{2}) \\ + m_{1} (- 12 θ m_{2} + (5 + 5 c + 12 θ C_{3}) n_{2}))) \end{matrix}$ ，

$M = \frac{18 θ^{4}}{625 c^{2} δ^{2}} - \frac{c^{2}}{2} - c - \frac{1}{2}$ ， $p (ξ) = C_{3} + \frac{θ}{10 c δ} \tanh (\frac{θ (ξ + 5 c δ C_{1})}{10 c δ})$ ，

$q (ξ) = C_{1} e^{C_{3} ξ + C_{2}} sech (\frac{θ (ξ + 5 c δ C_{1})}{10 c δ})$ ，

其中 $C_{i} (i = 1, 2, 3)$ 均为任意常数。将上面求得的 $p (ξ), q (ξ)$ 代入(5)式，有

，

其中 $C_{4}$ 均为任意常数。再由(7)式，我们得到了RLW-Burgers方程的解为

$\begin{matrix} u (ξ) = \frac{3 {(10 c δ m_{2} + (θ - 10 c δ C_{3}) n_{2})}^{2} {(- 10 c δ m_{2} + (θ + 10 c δ C_{3}) n_{2})}^{2}}{2500 c^{3} δ^{3} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} \\ \cdot \frac{{(10 c δ θ (C_{1} n_{1} e^{C_{2}} + C_{3} C_{4} n_{1} - C_{4} m_{1}) + (C_{4} θ^{2} n_{1} - 100 C_{1} c^{2} δ^{2} e^{C_{2}} (m_{1} - C_{3} n_{1})) \tanh (\frac{θ (ξ + 5 c δ C_{1})}{10 c δ}))}^{2}}{{(10 c δ θ (C_{1} n_{2} e^{C_{2}} + C_{3} C_{4} n_{2} - C_{4} m_{2}) + (C_{4} θ^{2} n_{2} - 100 C_{1} c^{2} δ^{2} e^{C_{2}} (m_{2} - C_{3} n_{2})) \tanh (\frac{θ (ξ + 5 c δ C_{1})}{10 c δ}))}^{2}} \\ + \frac{3 (10 c δ m_{1} - (θ + 10 c δ C_{3}) n_{1}) {(10 c δ m_{2} + (θ - 10 c δ C_{3}) n_{2})}^{2} (- 10 c δ m_{2} + (θ + 10 c δ C_{3}) n_{2})}{1250 c^{3} δ^{3} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} \end{matrix}$

$\begin{array}{l} \cdot \frac{10 c δ θ (C_{1} n_{1} e^{C_{2}} + C_{3} C_{4} n_{1} - C_{4} m_{1}) + (C_{4} θ^{2} n_{1} - 100 C_{1} c^{2} δ^{2} e^{C_{2}} (m_{1} - C_{3} n_{1})) \tanh (\frac{θ (ξ + 5 c δ C_{1})}{10 c δ})}{10 c δ θ (C_{1} n_{2} e^{C_{2}} + C_{3} C_{4} n_{2} - C_{4} m_{2}) + (C_{4} θ^{2} n_{2} - 100 C_{1} c^{2} δ^{2} e^{C_{2}} (m_{2} - C_{3} n_{2})) \tanh (\frac{θ (ξ + 5 c δ C_{1})}{10 c δ})} \\ + \frac{1}{2500 c^{3} δ^{3} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} (3 θ^{3} n_{1} n_{2} (20 c δ m_{2} n_{1} + (- 20 c δ m_{1} + θ n_{1}) n_{2}) \\ - 300 c^{2} δ^{2} θ^{2} (m_{2}^{2} n_{1}^{2} - 2 C_{3} m_{2} n_{1}^{2} n_{2} + (m_{1}^{2} - 2 C_{3} m_{1} n_{1} + 2 C_{3}^{2} n_{1}^{2}) n_{2}^{2}) \\ + 500 c^{3} δ^{3} (m_{2} n_{1} - m_{1} n_{2}) (- n_{1} ((5 + 5 c - 12 θ C_{3}) m_{2} + 12 θ C_{3}^{2} n_{2}) \\ + m_{1} (- 12 θ m_{2} + (5 + 5 c + 12 θ C_{3}) n_{2}))) \end{array}$

这里的 $ξ = x - c t$ 。

2) $a_{2} = \frac{245 θ n_{2} (m_{2} n_{1} - m_{1} n_{2})}{54 n_{1}^{3}}$ ， $a_{1} = \frac{70 θ (m_{2} n_{1} - m_{1} n_{2})}{27 n_{1}^{2}}$ ， $a_{0} = \frac{10 θ}{27} (\frac{m_{1}}{n_{1}} - \frac{m_{2}}{n_{2}}) - 1$ ，

$c = \frac{44}{135} θ (\frac{m_{1}}{n_{1}} - \frac{m_{2}}{n_{2}})$ ， $δ = - \frac{243 n_{1}^{2} n_{2}^{2}}{352 {(m_{2} n_{1} - m_{1} n_{2})}^{2}}$ ，

$M = \frac{3760 θ^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2} + 2376 θ n_{1} n_{2} (m_{2} n_{1} - m_{1} n_{2}) - 3645 n_{1}^{2} n_{2}^{2}}{7290 n_{1}^{2} n_{2}^{2}}$ ，

$p (ξ) = \frac{(- 2 m_{2} n_{1} + 5 m_{1} n_{2}) e^{\frac{8 m_{1}}{9 n_{1}} ξ + 24 C_{1} m_{2} n_{1}^{2} n_{2}} + (2 m_{2} n_{1} + 7 m_{1} n_{2}) e^{\frac{8 m_{2}}{9 n_{2}} ξ + 24 C_{1} m_{1} n_{1} n_{2}^{2}}}{3 n_{1} n_{2} e^{\frac{8 m_{1}}{9 n_{1}} ξ + 24 C_{1} m_{2} n_{1}^{2} n_{2}} + 9 n_{1} n_{2} e^{\frac{8 m_{2}}{9 n_{2}} ξ + 24 C_{1} m_{1} n_{1} n_{2}^{2}}}$ ，

$q (ξ) = \frac{C_{2} e^{(\frac{5 m_{1}}{3 n_{1}} + \frac{2 m_{2}}{9 n_{2}}) ξ}}{e^{\frac{8 m_{1}}{9 n_{1}} ξ + 24 C_{1} m_{2} n_{1}^{2} n_{2}} + 3 e^{\frac{8 m_{2}}{9 n_{2}} ξ + 24 C_{1} m_{1} n_{1} n_{2}^{2}}}$ ，

其中 $C_{i} (i = 1, 2, 3)$ 均为任意常数。将上面求得的 $p (ξ), q (ξ)$ 代入(5)式，有

$G (ξ) = \frac{8 (m_{2} n_{1} - m_{1} n_{2}) e^{\frac{1}{9} (- \frac{7 m_{1}}{n_{1}} + \frac{6 m_{2}}{n_{2}}) ξ}}{9 C_{2} n_{1} n_{2} e^{\frac{8 m_{2}}{9 n_{2}} (ξ - 27 C_{1} n_{1}^{2} n_{2}^{2})} + 8 C_{3} (m_{2} n_{1} - m_{1} n_{2}) (e^{\frac{8 m_{1}}{9 n_{1}} ξ + 24 C_{1} m_{2} n_{1}^{2} n_{2}} + 3 e^{\frac{8 m_{2}}{9 n_{2}} ξ + 24 C_{1} m_{1} n_{1} n_{2}^{2}})}$ ，

再由(7)式，我们得到了RLW-Burgers方程的解为

$\begin{matrix} u (ξ) = \frac{490 θ (m_{2} n_{1} - m_{1} n_{2}) {(3 C_{2} n_{1} n_{2} e^{\frac{8 m_{2}}{9 n_{2}} ξ} - 8 C_{3} (m_{2} n_{1} - m_{1} n_{2}) (e^{\frac{8 m_{1}}{9 n_{1}} ξ + 48 C_{1} m_{2} n_{1}^{2} n_{2}} - e^{\frac{8 m_{2}}{9 n_{2}} ξ + 24 C_{1} n_{1} n_{2} (m_{2} n_{1} + m_{1} n_{2})}))}^{2}}{27 n_{1} n_{2} {(21 C_{2} n_{1} n_{2} e^{\frac{8 m_{2}}{9 n_{2}} ξ} + 8 C_{3} (m_{2} n_{1} - m_{1} n_{2}) ({5e}^{\frac{8 m_{1}}{9 n_{1}} ξ + 48 C_{1} m_{2} n_{1}^{2} n_{2}} + 7 e^{\frac{8 m_{2}}{9 n_{2}} ξ + 24 C_{1} n_{1} n_{2} (m_{2} n_{1} + m_{1} n_{2})}))}^{2}} \\ - \frac{140 θ (m_{2} n_{1} - m_{1} n_{2}) (3 C_{2} n_{1} n_{2} e^{\frac{8 m_{2}}{9 n_{2}} ξ} - 8 C_{3} (m_{2} n_{1} - m_{1} n_{2}) (e^{\frac{8 m_{1}}{9 n_{1}} ξ + 48 C_{1} m_{2} n_{1}^{2} n_{2}} - e^{\frac{8 m_{2}}{9 n_{2}} ξ + 24 C_{1} n_{1} n_{2} (m_{2} n_{1} + m_{1} n_{2})}))}{27 n_{1} n_{2} (21 C_{2} n_{1} n_{2} e^{\frac{8 m_{2}}{9 n_{2}} ξ} + 8 C_{3} (m_{2} n_{1} - m_{1} n_{2}) ({5e}^{\frac{8 m_{1}}{9 n_{1}} ξ + 48 C_{1} m_{2} n_{1}^{2} n_{2}} + 7 e^{\frac{8 m_{2}}{9 n_{2}} ξ + 24 C_{1} n_{1} n_{2} (m_{2} n_{1} + m_{1} n_{2})}))} \\ + \frac{10 θ}{27} (\frac{m_{1}}{n_{1}} - \frac{m_{2}}{n_{2}}) - 1 \end{matrix}$ ，

这里的 $ξ = x - \frac{44}{135} θ (\frac{m_{1}}{n_{1}} - \frac{m_{2}}{n_{2}}) t$ 。

3) $a_{2} = \frac{12 {(m_{2} - C_{1} n_{2})}^{2} {(5 c δ (m_{2} - C_{1} n_{2}) \mp θ n_{2})}^{2}}{25 c δ {(m_{2} n_{1} - m_{1} n_{2})}^{2}}$ ，

$\begin{matrix} a_{1} = \frac{6}{125 c^{2} δ^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} (10 c δ (m_{1} - C_{1} n_{1}) - (1 \pm 1) θ n_{1}) (m_{2} - C_{1} n_{2}) \\ \times (5 c δ (m_{2} - C_{1} n_{2}) \mp θ n_{2}) (10 c δ (m_{2} - C_{1} n_{2}) + (1 \mp 1) θ n_{2}) \end{matrix}$ ，

$\begin{matrix} a_{0} = \frac{1}{125 c^{2} δ^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} (n_{1}^{2} (5 c δ (25 c^{2} δ (12 C_{1}^{2} δ - 1) \pm 6 θ^{2} + 5 c δ (- 5 + 12 C_{1} θ (1 \pm 1))) m_{2}^{2} \\ - 12 {(5 c δ C_{1} + θ)}^{2} (10 c δ C_{1} + (- 1 \pm 1) θ) m_{2} n_{2} + 60 c δ C_{1}^{2} {(5 c δ C_{1} \pm θ)}^{2} n_{2}^{2}) \\ - m_{1} n_{1} (300 c^{2} δ^{2} (10 c δ C_{1} + (1 \pm 1) θ) m_{2}^{2} - 10 c δ (25 c^{2} δ (1 + 24 C_{1}^{2} δ) + 6 θ^{2} + 5 c δ (5 \pm 24 C_{1} θ)) m_{2} n_{2} \\ + 3 {(10 c δ C_{1} + (- 1 \pm 1) θ)}^{2} (10 c δ C_{1} + (1 \pm 1) θ) n_{2}^{2}) + 5 c δ m_{1}^{2} (300 c^{2} δ^{2} m_{2}^{2} \\ - 60 c δ (10 c δ C_{1} + (- 1 \pm 1) θ) m_{2} n_{2} + (25 c^{2} δ (12 δ C_{1}^{2} - 1) + 5 c δ (- 5 + 12 θ C_{1} (- 1 \pm 1)) \mp 6 θ^{2}) n_{2}^{2})) \end{matrix}$ ，

$M = \frac{18 θ^{4}}{625 c^{2} δ^{2}} - \frac{c^{2}}{2} - c - \frac{1}{2}$ ， $p (ξ) = C_{1} \pm \frac{θ}{5 c δ}$ ， $q (ξ) = C_{2} e^{C_{1} ξ}$ ，

其中 $C_{i} (i = 1, 2, 3)$ 均为任意常数。将上面求得的 $p (ξ), q (ξ)$ 代入(5)式，有

$G (ξ) = \frac{θ e^{- C_{1} ξ}}{C_{3} θ e^{\pm \frac{θ}{5 c δ} ξ} \mp 5 c δ C_{2}}$ ，

再由(7)式，我们得到了RLW-Burgers方程的解为

$u (ξ) = \frac{12 {(m_{2} - C_{1} n_{2})}^{2}}{25 c δ {(m_{2} n_{1} - m_{1} n_{2})}^{2}} \frac{{(C_{3} θ e^{^{^{\pm \frac{θ}{5 c δ} ξ}}} (- 5 c δ (m_{1} - C_{1} n_{1}) \pm θ n_{1}) \pm 25 c^{2} δ^{2} C_{2} (m_{1} - C_{1} n_{1}))}^{2}}{{(C_{3} θ e^{^{\pm \frac{θ}{5 c δ} ξ}} (- 5 c δ (m_{2} - C_{1} n_{2}) \pm θ n_{2}) \pm 25 c^{2} δ^{2} C_{2} (m_{2} - C_{1} n_{2}))}^{2}}$

$\begin{matrix} \times {(5 c δ (m_{2} - C_{1} n_{2}) \mp θ n_{2})}^{2} + \frac{6}{125 c^{2} δ^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} (10 c δ (m_{1} - C_{1} n_{1}) - (1 \pm 1) θ n_{1}) \\ \times (m_{2} - C_{1} n_{2}) (5 c δ (m_{2} - C_{1} n_{2}) \mp θ n_{2}) (10 c δ (m_{2} - C_{1} n_{2}) + (1 \mp 1) θ n_{2}) \\ \times \frac{(C_{3} θ e^{^{\pm \frac{θ}{5 c δ} ξ}} (- 5 c δ (m_{1} - C_{1} n_{1}) \pm θ n_{1}) \pm 25 c^{2} δ^{2} C_{2} (m_{1} - C_{1} n_{1}))}{(C_{3} θ e^{^{^{\pm \frac{θ}{5 c δ} ξ}}} (- 5 c δ (m_{2} - C_{1} n_{2}) \pm θ n_{2}) \pm 25 c^{2} δ^{2} C_{2} (m_{2} - C_{1} n_{2}))} \\ + \frac{1}{125 c^{2} δ^{2} {(m_{2} n_{1} - m_{1} n_{2})}^{2}} (n_{1}^{2} (5 c δ (25 c^{2} δ (12 C_{1}^{2} δ - 1) \pm 6 θ^{2} + 5 c δ (- 5 + 12 C_{1} θ (1 \pm 1))) m_{2}^{2} \end{matrix}$

$\begin{matrix} - 12 {(5 c δ C_{1} + θ)}^{2} (10 c δ C_{1} + (- 1 \pm 1) θ) m_{2} n_{2} + 60 c δ C_{1}^{2} {(5 c δ C_{1} \pm θ)}^{2} n_{2}^{2}) \\ - m_{1} n_{1} (300 c^{2} δ^{2} (10 c δ C_{1} + (1 \pm 1) θ) m_{2}^{2} - 10 c δ (25 c^{2} δ (1 + 24 C_{1}^{2} δ) + 6 θ^{2} \\ + 5 c δ (5 \pm 24 C_{1} θ)) m_{2} n_{2} + 3 {(10 c δ C_{1} + (- 1 \pm 1) θ)}^{2} (10 c δ C_{1} + (1 \pm 1) θ) n_{2}^{2}) \\ + 5 c δ m_{1}^{2} (300 c^{2} δ^{2} m_{2}^{2} - 60 c δ (10 c δ C_{1} + (- 1 \pm 1) θ) m_{2} n_{2} \\ + (25 c^{2} δ (12 δ C_{1}^{2} - 1) + 5 c δ (- 5 + 12 θ C_{1} (- 1 \pm 1)) \mp 6 θ^{2}) n_{2}^{2})) \end{matrix}$

这里的 $ξ = x - c t$ 。

4) $a_{2} = - \frac{2 {(2 θ (m_{2} n_{1} - m_{1} n_{2}) - n_{1} n_{2})}^{2}}{5 n_{1}^{4}}$ ， $a_{1} = 0$ ， $a_{0} = \frac{2 (θ^{2} - δ)}{15 δ}$ ， $c = - \frac{2 θ^{2}}{15 δ}$ ， $M = 0$ ，

$p (ξ) = \frac{(2 θ m_{1} + n_{1}) e^{\frac{ξ}{2 θ}} - m_{1} e^{C_{1} n_{1}^{2}}}{2 n_{1} θ e^{\frac{ξ}{2 θ}} - n_{1} e^{C_{1} n_{1}^{2}}}$ ， $q (ξ) = \frac{C_{2} e^{(\frac{1}{2 θ} + \frac{m_{1}}{n_{1}}) ξ}}{- 2 θ e^{\frac{ξ}{2 θ}} + e^{C_{1} n_{1}^{2}}}$ ，

其中 $C_{i} (i = 1, 2, 3)$ 均为任意常数。将上面求得的 $p (ξ), q (ξ)$ 代入(5)式，有

$G (ξ) = \frac{e^{- \frac{m_{1}}{n_{1}} ξ}}{C_{2} + C_{3} e^{C_{1} n_{1}^{2}} - 2 θ C_{3} e^{\frac{ξ}{2 θ}}}$ ，

再由(7)式，我们得到了RLW-Burgers方程的解为

$u (ξ) = - \frac{2 C_{3}^{2} {(2 θ (m_{2} n_{1} - m_{1} n_{2}) - n_{1} n_{2})}^{2} e^{\frac{ξ}{θ}}}{5 {((C_{2} + C_{3} e^{C_{1} n_{1}^{2}}) (m_{2} n_{1} - m_{1} n_{2}) - C_{3} (2 θ (m_{2} n_{1} - m_{1} n_{2}) - n_{1} n_{2}) e^{\frac{ξ}{2 θ}})}^{2}} + \frac{2 (θ^{2} - δ)}{15 δ}$ ，

这里的 $ξ = x + \frac{2 θ^{2}}{15 δ} t$ 。

5) $a_{2} = \frac{2 {(2 θ (m_{2} n_{1} - m_{1} n_{2}) + n_{1} n_{2})}^{2}}{5 n_{1}^{4}}$ ， $a_{1} = 0$ ， $a_{0} = - \frac{2 (θ^{2} + 14 δ)}{15 δ}$ ， $c = \frac{2 θ^{2}}{15 δ}$ ， $M = 0$ ，

$p (ξ) = \frac{m_{1} e^{\frac{ξ}{2 θ}} - (2 θ m_{1} - n_{1}) e^{C_{1} n_{1}^{2}}}{n_{1} e^{\frac{ξ}{2 θ}} - 2 n_{1} θ e^{C_{1} n_{1}^{2}}}$ ， $q (ξ) = \frac{C_{2} e^{\frac{m_{1}}{n_{1}} ξ}}{e^{\frac{ξ}{2 θ}} - 2 θ e^{C_{1} n_{1}^{2}}}$ ，

其中 $C_{i} (i = 1, 2, 3)$ 均为任意常数。将上面求得的 $p (ξ), q (ξ)$ 代入(5)式，有

$G (ξ) = \frac{e^{\frac{1}{2} (\frac{1}{θ} - \frac{2 m_{1}}{n_{1}}) ξ}}{C_{3} e^{\frac{ξ}{2 θ}} - 2 θ C_{3} e^{C_{1} n_{1}^{2}} - 2 θ C_{2}}$ ，

再由(7)式，我们得到了RLW-Burgers方程的解为

$u (ξ) = \frac{2 {(2 θ (m_{2} n_{1} - m_{1} n_{2}) + n_{1} n_{2})}^{2} {(C_{2} + C_{3} e^{C_{1} n_{1}^{2}})}^{2}}{5 {((2 θ (m_{2} n_{1} - m_{1} n_{2}) + n_{1} n_{2}) (C_{2} + C_{3} e^{C_{1} n_{1}^{2}}) - C_{3} (m_{2} n_{1} - m_{1} n_{2}) e^{\frac{ξ}{2 θ}})}^{2}} - \frac{2 (θ^{2} + 14 δ)}{15 δ}$ ，

这里的 $ξ = x - \frac{2 θ^{2}}{15 δ} t$ 。

4. 总结

本文构造了一类满足变系数Bernoulli方程的，且含有自由参数 $m_{i}, n_{i} (i = 1, 2)$ 的 $(\frac{m_{1} G + n_{1} G^{'}}{m_{2} G + n_{2} G^{'}})$ 展开法。用此方法对RLW-Burgers方程进行了求解，得到了该方程的多个新的显式行波解。事实证明，该方法不仅可以求解非线性偏微分方程，而且可以得到方程更为丰富的精确解。

基金项目

国家自然科学基金(11601109)，海南省自然科学基金(117066)。

文章引用

王鑫. 满足变系数方程的G展开法及RLW-Burgers方程的新精确解
The G Expansion Method of Satisfying a Variable Coefficient Equation and New Exact Solutions of Rlw-Burgers Equation[J]. 应用数学进展, 2018, 07(01): 80-90. http://dx.doi.org/10.12677/AAM.2018.71010

参考文献 (References)

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