推广的KdV方程特征问题解的存在唯一性

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

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 generalized KDV equation, u_t+αuu_x+μu_x3+εu_x5=0, is a typical integrable equation. It is derived by studying the disseminations of magnet sound wave in cold plasma[2], the isolated wave in transmission line￣[3], and the isolated wave in the boundary surface of the divided layer fluid [4]. For the characteristic problem of the generalized KdV equation, this paper, on the basis of Riemann function, designs a suitable structure, then changes the characteristic problem into an equivalent integral and differential equation whose corresponding mapping E is a mapping to itself [5]. According to the principle of fixed point, the above integral and differential equation has a unique regular solution, so the character-istic problem of the generalized KdV equation has a unique solution. The itera-tive solution derived from the integral-differential equation sequence is uniformly convergent in Ω.