求解高阶KdV方程孤子解的一种简单方法

求解高阶KdV方程孤子解的一种简单方法 A Simple Method for Solving the Higher Order KdV Equation 何进春 HE Jinchun1 1 first-author 陈化 CHEN Hua2 2 胡适耕 HU Shigeng1 1 华中科技大学数学系 华中科技大学数学系 Department of Mathematics, Huazhong University of Science and Technology Department of Mathematics, Huazhong University of Science and Technology 武汉大学数学与统计学院 武汉大学数学与统计学院 School of Mathematics and Statistics, Wuhan University School of Mathematics and Statistics, Wuhan University 根据Jost解必须满足相容性方程,从2n+1阶KdV方程的第一个相容性方程推导出两个等式;用这两个等式构造出Gel’fand-Levitan-Marchenko方程,在无反射条件下,求解任意高阶KdV方程孤子解的问题归结为纯粹的代数运算,从而避开了对Jost解的解析性进行繁杂的理论分析,因此更加便于实际应用. The soliton solution to the KdV equation with 3rd and 5th order obtained many years ago by the inverse scattering method. But it was complicated to analyze the analyticity of Jost solutions. In the paper, we constructed the Gel’fand-Levitan-Marchenko equation through the conditions deduced from the fact that the Jost solutions satisfied the two compatibility equations. Using the simple method, in the non-scattering case, the problem of solving the soliton solution to the KdV equation with 2n+1 order deduced to the algebraic calculation. Therefore, the analyticity of Jost solutions and the explicit expression of the 2n+1 order KdV equation were unnecessary. 高阶KdV方程 反散射法 Gel’fand-Levitan-Marchenko方程 higher order KdV equation inverse scattering method Gel’fand-Levitan-Marchenko equation O175.2 华中科技大学博士后专项基金资助项目（0101011110） 2008-01-01 2021-04-01 1