(1+1)维广义的浅水波方程的变量分离解和孤子激发模式

研究(1+1)维广义的浅水波方程的变量分离解和孤子激发模式. 该方程包括两种完全可积(IST可积)的特殊情况，分别为AKNS方程和Hirota-Satsuma方程. 首先把基于Bcklund变换的变量分离(BT-VS)方法推广到该方程，得到了含有低维任意函数的变量分离解. 对于可积的情况，含有一个空间任意函数和一个时间任意函数，而对于不可积的情况，仅含有一个时间任意函数，其空间函数需要满足附加条件. 另外，对于得到的(1+1)维普适公式，选取合适的函数，构造了丰富的孤子激发模式，包括单孤子，正-反孤子，孤子膨胀，类呼吸子，类瞬子等等. 最后，对BT-VS方法作一些讨论. 关键词： 浅水波方程 Bcklund变换 变量分离 孤子

Variable separation solution and soliton excitations of the (1+1)-dimensional generalised shallow water wave equation

In this paper, variable separation solution and soliton excitations of the (1+1)-dimensional generalised shallow water wave equation are obtained. This equation includes two special cases which are completely integrable (IST integrable): the AKNS equation and the Hirota-Satsuma equation. Firstly, the variable separation (BT-VS) method based on the Bcklund transformation is extended to this eqaution for deriving VS solutions which include some low dimensional arbitrary functions. In the integrable cases, a space arbitrary function and a time arbitrary function are included. But in the other cases only a time arbitrary function is included and the space function needs to satisfy a specific condition. In addition, for the (1+1)-dimensional universal formula, abundant soliton excitations can be constructed, such as one-soliton, bell-anti-bell soltion, soliton expansion, breather-like, instaton-like. Finally, some discusions are made about the VT-VS method.