修正的 VN 方程组的对称、约化和精确解

摘要

To get new exact solutions of the mVN system,a relationship was constructed between the new solutions and the old ones of the mVN system,by using an improved direct CK method.Based on the relationship,new solutions were obtained by u-sing a given solution of the equation.At the same time,the symmetry of the mVN system was also obtained.By solving the re-duction equation,some kinds of explicit solutions of the mVN system were obtained,including the power series solution,the Air-y function solution and Jacobi elliptic function solution.The results show that the improved direct CK method is effective to solve nonlinear evolution equations.%为了得到修正的 VN 方程组的精确解，利用改进的 CK 直接方法研究该方程组，建立了该方程组新、旧解之间的关系，基于此关系推广了方程组的解；同时，得到了该方程组的对称，通过求解约化方程，得到修正的 VN 方程组的许多新的精确解，包括幂级数解、艾米尔函数解、雅克比椭圆函数解等。结果表明，改进的 CK 直接方法对于求解此类非线性发展方程的精确解是有效的。