基于Bell多项式的一类(3+1)维变系数广义浅水波方程的可积性研究  

INTEGRABILITY OF A VARIABLE-COEFFICIENT GENERALIZED(3+1)DIMENSIONAL SHALLOW WATER WAVE EQUATION VIA BELL POLYNOMIALS

作　　者：李春晖 王丹[1] 刘淑丽 李金红 王晓丽[1] LI Chun-hui;WANG Dan;LIU Shu-li;LI Jin-hong;WANG Xiao-li(School of Mathematics and Statistics,Qilu University of Technology(Shandong Academy of Sciences),ShandongJinan250353)

机构地区：[1]齐鲁工业大学(山东省科学院)数学与统计学院,山东济南250353

出　　处：《数学杂志》2023年第6期487-500,共14页Journal of Mathematics

基　　金：国家自然科学基金资助(11801292);山东省自然科学基金资助(ZR2020MA049)。

摘　　要：本文基于Bell多项式研究了一类(3+1)维变系数广义浅水波方程的可积性问题.首先,引入变量变换,借助Bell多项式与Hirota双线性算子之间的关系,导出方程的Hirota双线性形式,求出方程的N-孤子解,并对单孤子、双孤子和三孤子在不同情形下的传播进行图像模拟;其次,基于双线性方程,结合Bell多项式获得方程的双线性Backlund变换;然后,通过Hopf-Cole变换,将双线性Backlund变换线性化,求出方程的Lax对;最后,利用级数展开法得到方程的无穷守恒律.从而证明该方程具有可积性.In this paper,we focus on a(3+1)dimensional variable-coeficient generalized shallow water wave equation based on Bell polynomials.Firstly,the transformation of variables is introduced,and the Hirota bilinear form of the equation is derived by the relation between Bell polynomial and the Hirota bilinear operator.The N-soliton solution of the equation is obtained,and the propagation of single soliton,double soliton and triple soliton in different cases are simulated.Furthermore,the bilinear Backlund transformation is obtained based on the bilinear equation and Bell polynomials.Then,through the Hopf-Cole transformation,the bilinear Backlund transformation is linearized,and the Lax pair of the equation is obtained.Finally,the infinite conservation law of the equation is obtained by using the series expansion method.Thus,the integrability of the equation is proved.

关 键 词：广义浅水波方程 BELL多项式 B?cklund变换 LAX对 无穷守恒律 

分 类 号：O175.29[理学—数学]