非线性麦克斯韦方程最优系统及精确解  

Optimal Systems and Exact Solutions of a Nonlinear Maxwell Equation

作　　者：郭增鑫 胡彦鑫 辛祥鹏[1] GUO Zengxin;HU Yanxin;XIN Xiangpeng(School of Mathematical Sciences,Liaocheng University,Liaocheng 252059,China)

机构地区：[1]聊城大学数学科学学院,山东聊城252059

出　　处：《聊城大学学报（自然科学版）》2021年第4期15-22,53,共9页Journal of Liaocheng University:Natural Science Edition

基　　金：国家自然科学基金项目(11505090)资助。

摘　　要：研究一类二阶非线性麦克斯韦方程的对称约化以及精确解问题。首先利用李群方法求出该方程的向量场,进而方程的对称也可以得到,并通过求解常微分方程初值问题得到了该方程的对称群。其次,为了研究对称的等价性,利用一维最优化方法得到该方程的最优系统,借助最优系统对方程进行对称约化。为了方便求解约化后的常微分方程,对一些参数做了一定的约束,因此得到的解是在上述约束条件下的解析解。最后构造了这些情况下的群不变解,并求出一系列新的精确解。Study the symmetry reductions and exact solutions of a class of second-order nonlinear Maxwell equation.Firstly,under the constraint of invariable form,the vector fields and symmetries of the equation are obtained by using Lie group method.Secondly,in order to study the equivalence of symmetry,the one-dimensional optimization method is used to obtain the optimal system of the equation,and the equation is reduced with the help of the optimal system.In order to solve the reduced ordinary differential equations,some parameters are constrained,so solutions are the analytic solutions under the above constraints.Finally,the group invariant solutions in these cases are constructed,and a series of new exact solutions are obtained.

关 键 词：非线性麦克斯韦方程 李群 对称 精确解 最优系统 

分 类 号：O175[理学—数学]