五阶KdV方程的李对称分析、对称约化以及解析解  

Lie Symmetry Analysis,Symmetry Reductions and Analytical Solution of the Fifth-order KdV Equation

作　　者：秦春艳 QIN Chunyan(School of Mathematics and Statistics,Suzhou University,Suzhou 234000,China)

机构地区：[1]宿州学院数学与统计学院,安徽宿州234000

出　　处：《宿州学院学报》2023年第6期6-11,共6页Journal of Suzhou University

基　　金：安徽省教育厅高校自然科学研究项目(KJ2021ZD0136,KJ2021A1102);宿州学院重点科研项目(2020yzd06);宿州学院博士科研启动基金(2020BS011)。

摘　　要：非线性偏微分方程是现代数学的一个重要分支,研究它的精确解析解具有重要理论意义。而李对称分析法是求解非线性偏微分方程的一种有效工具。KdV方程是可积系统中经典的数学模型,在当前许多科学和工程领域的理论研究中具有非常重要意义。五阶KdV方程可以用来模拟激光光学和等离子体物理等科学领域的非线性色散波。首先,通过李对称分析法得到李点对称和优化系统;其次,基于优化系统获得对称约化和群不变解;再次,利用幂级数法构造精确解析解,进而对所求精确幂级数解进行收敛性分析;最后,利用孤子拟设法,求出五阶KdV方程的奇异孤立波解,并且通过选取适当的参数,借助Maple数学软件绘制了幂级数解和奇异孤立波解的相关图形,这些结果可以丰富非线性动力系统的行为。Nonlinear partial differential equation is an important branch of modern mathematics,and it is of great theoretical significance to study its exact analytic solution.The Lie symmetry analysis method is a useful tool for solving nonlinear partial differential equations.KdV equation is a classical mathematical model of integrable system,which plays an important role in many theoretical researches in science and engineering fields.The fifth-order KdV equation can be used to model nonlinear dispersive waves in scientific areas such as laser optics and plasma phy-sics.Firstly,the Lie symmetry analysis method is performed to get the Lie point symmetries and optimal systems,respectively.Then the symmetry reductions and group invariant solutions are obtained based on the optimal systems.Secondly,the power series method is used to construct exact analytic solution.Furthermore,convergence analysis of the exact power series solution for fifth-order KdV are presented.Finally,using the solitary ansatz method,the singular solitary wave solution of the fifth-order KdV equation is obtained.By selecting the appropriate parameters,with the aid of mathematical software Maple,the relevant graphics of the power series solution and singular solitary wave solution are mapped.These results can enrich the behavior of the nonlinear dynamic system.

关 键 词：五阶KDV方程 李对称性分析 对称约化 幂级数法 精确解析解 收敛性分析 孤子拟设法 

分 类 号：O175[理学—数学]