Kadomtsev-Petviashvilli方程的直接相似约化

Clarkson和Kruskal发展的直接法(CK直接法)是求解非线性微分方程相似约化的一种强有力的方法. 本文以Kadomtsev-Petviashvilli(KP)方程为例, 运用CK直接法把KP方程简化为3种类型的(1+1)维偏微分方程, 这3种偏微分方程等价于经典Lie方法得到的3种具有不同独立变量的相似约化方程. KP方程的解包含了更多经典Lie方法所遗漏的任意函数, 例如, CK直接法得到的第3类约化可以分为3个子情形, 而经典Lie法得到的KP方程的第3类解只是我们结果的一个子情形的特例.

Similarity reductions of the KP equation using a direct method

The direct method developed by Clarkson and Kruskal is a very efficient method to solve the similarity reduction of nonlinear differential equations. In this paper the Kadomtsev-Petviashvilli (KP) equation is taken as an example to simplify KP equation into three types of (1+1) dimensional partial differential equations using CK direct method. These three partial differential equations are equivalent to the three similar reduction equations with different independent variables obtained through the classical Lie method. The solution to the KP equation contains more arbitrary functions which the classical Lie method lacks. For example, the third type of reduction obtained using the direct method can be divided into three subcases, while the third type of solution to the KP equation obtained using the classical Lie method serves as a special case of a subcase of the result derived in this work.