推广的KdV方程特征问题解的存在唯一性

推广的KdV方程u_t+αuu_x+μu_x3+εu_x5=0[1]是典型的可积方程.它先后在研究冷等离子体中磁声波的传播[2],传输线中孤立波[3]和分层流体中界面孤立波[4]时导出.本文对推广的KdV方程的特征问题,在Riemann函数的基础上,设计一恰当结构,并由此化待征问题为一与之等价的积分微分方程.而该积分微分方程对应的映射E是列自身的映射[5],依不动点原理,积分微分方程有唯一的正则解,即推广的KdV方程的特征问题有唯一解,且由积分微分方程序列所得的迭代解于Ω上一致收敛.

The Uniqueness and Existence of Solution of the Charac-teristic Problem on the Generalized KdV Equation

The generalized KdV equation u_t+αuu_x+μu_x3+εu_x5=0[1] is a typical integr-able equation.It is derived studying the dissemination of magnet sound wave in coldplasma[2],Ihe isolated wave in transmission line[3],and the isolated wave in the bound-ary surface of the divided layer fluid[4].For the characteristic problem of the gene-ralized KdV equation,this paper,based on the Riemann function,designs a suitable structure,then changes the characteristic problem to an equivalent integral and dif-ferential equation whose corresponding mapping E is a mapping to itself[5].According to the principle of fixed point,the above integral differential equ-ation has a unique regular solution,so the characteristic problem of the generalized KdV equation has a.unique solution.The iteration solution derived from the integral differential equation sequence is uniformly convegent in Ω.