## 一类非线性动力系统的孤立子波解

1 湖州师范学院理学院 浙江湖州 313000

2 安徽师范大学数学与统计学院 安徽芜湖 241003

## The Solitary Wave Solution to a Class of Nonlinear Dynamic System

Ouyang Cheng1, Mo Jiaqi,2

1 Faculty of Science, Huzhou University, Zhejiang Huzhou 313000

2 School of Mathematics and Statistic, Anhui Normal University, Anhui Wuhu 241003

 基金资助: 国家自然科学基金.  11771005浙江省自然科学基金.  LY13A010005浙江省自然科学基金.  KJ2018A0964浙江省自然科学基金.  KJ2017A901

 Fund supported: the NSFC.  11771005the NSF of Zhejiang Province.  LY13A010005the NSF of Zhejiang Province.  KJ2018A0964the NSF of Zhejiang Province.  KJ2017A901

Abstract

Using the functional generalized variational iteration method, a class of nonlinear disturbed dynamic system was considered. First introduce solitary solution to a corresponding typical system. And then a set of functional generalized variation constructed, and Lagrange multiplier functions were solved. Finally, the generalized variational iteration was received. Thus, the asymptotic travelling wave solution to the original nonlinear disturbed generalized dynamic system was obtained

Keywords： Dynamic system ; Nonlinear ; Solitary wave

Ouyang Cheng, Mo Jiaqi. The Solitary Wave Solution to a Class of Nonlinear Dynamic System. Acta Mathematica Scientia[J], 2021, 41(6): 1830-1837 doi:

## 1 引言

\begin{align} p_{t}+p_{xxx}-a_{1}pq_{x}-a_{2}qp_{y} = \varepsilon F_{1}(p, q), \end{align}

\begin{align} p_{x}-a_{3}q_{y} = \varepsilon F_{2}(p, q). \end{align}

\begin{align} z = b_{1}x+b_{2}y+b_{1}t, \end{align}

\begin{align} b^{3}_{1}p_{zzz}-b_{3}p_{z}-a_{1}b_{1}pq_{z}-a_{2}b_{1}qp_{z} = \varepsilon F_{1}(p, q), \end{align}

\begin{align} b_{1}p_{z}-a_{3}b_{1}q_{z} = \varepsilon F_{2}(p, q). \end{align}

## 2 典型非线性动力系统模型

\begin{align} b^{3}_{1}p_{zz}-b_{3}p_{z}-a_{1}b_{1}pq_{z}-a_{2}b_{2}qp_{z} = 0, \end{align}

\begin{align} b_{1}p_{z}-a_{3}b_{2}q_{z} = 0. \end{align}

\begin{align} b^{3}_{1}p_{zz}-b_{3}p-\frac{b_{1}(a_{1}b_{1}+a_{2}b_{2})}{2a_{3}b_{2}}p^{2} = 0, \end{align}

\begin{align} q = \frac{b_{1}}{a_{3}b_{2}}p. \end{align}

\begin{align} p(z) = A_{1}{\rm sech}\ z+A_{2}{\rm sech}^{2}z+B_{0}\tanh z, \end{align}

\begin{align} p_{z} = -A_{1}{\rm sech}\ z\tanh z-2A_{2}{\rm sech}^{2}z\tanh z+B_{0}\tanh\ z, \end{align}

\begin{align} p_{zz} = A_{1}{\rm sech}\ z+4A_{2}{\rm sech}^{2}z-2A_{1}{\rm sech}^{3}z-6A_{2}{\rm sech}^{4}z -2B_{0}{\rm sech}^{2}z\tanh z. \end{align}

\begin{align} z = b_{1}x+b_{2}y-4b^{3}_{1}t. \end{align}

\begin{align} U(z) = -\frac{12a_{3}b^{2}_{1}b_{2}}{a_{1}b_{1}+a_{2}b_{2}}{\rm sech}^{2}z, \end{align}

\begin{align} V(z) = -\frac{12b^{3}_{1}}{a_{1}b_{1}+a_{2}b_{2}}{\rm sech}^{2}z. \end{align}

## 4 例子

$$$p_z(t)+p_{zzz}+pq_{z}-qp_{z} = \varepsilon\sin p,$$$

$$$-p_{z}-3q_{z} = \varepsilon\cos q,$$$

$$$p_{0}(z) = -36{\rm sech}^{2}z,$$$

$$$q_{0}(z) = -12{\rm sech}^{2}z.$$$

$$$p_{1}(z) = -36{\rm sech}^{2}z +\varepsilon\int^{z}_{-\infty}\bigg[\sum^{3}\limits_{i = 1} C_{i}\exp(\delta_{i}r)\bigg]\sin(-36{\rm sech}^{2}r){\rm d}r-\varepsilon\sin p_{0},$$$

$$$q_{1}(z) = -12{\rm sech}^{2}z+\varepsilon\int^{z}_{-\infty}\int^{z}_{-\infty} [\cos(12{\rm sech}^{2}r)]{\rm d}r-\varepsilon\cos q_{0},$$$

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