基于符号计算的变系数KP方程的孤子解

摘要

Computer algebra is the interdisciplinary subject of mathematics, computer science and artificial intelligence, studying symbolic computation on computer.Maple, as the most widely used symbolic computation system in the world, can turn complex computational process into algorithms issued by computers, and hand it over to the computer for processing, which greatly improves the calculation efficiency.In this paper, with the symbolic calculation system Maple as the working platform, the modified CK direct reduction method is adopted to reduce the KP equation with variable coefficients to the equivalent constant coefficient equation, and the relationship and solution between the variable coefficient and the constant coefficient equation are obtained.Painlevé nonstandard truncation method is extended to obtain the solution of the corresponding ordinary differential equation, then getting the solution of the differential equation with variable coefficients.The soliton solution of KP equation with constant coefficients is analyzed.Studies show that the combination of modified CK direct reduction method and Painlevé non-standard truncation method to calculate Maple system with the help of sign can not only solve partial differential equation with variable coefficient easily, but also save computation time, which is widely used in several mathematical and physical models.%计算机代数是数学和计算机科学以及人工智能的交叉学科, 致力于研究在计算机上进行符号计算.Maple是世界上应用最广泛的符号计算系统, 可以将复杂繁琐的计算过程算法化, 交由计算机处理, 将人类的双手解放出来, 极大地提高了计算效率.文中以符号计算系统Maple为工作平台, 采用修正的CK直接约化方法, 将变系数KP方程约化为等价常系数方程, 获得到了变系数与常系数方程的关系及解.通过推广的Painlevé非标准截断方法求得相应的常微分方程的解, 进而得到变系数微分方程的解, 分析了常系数KP方程的孤子解.研究表明, 借助符号计算Maple系统, 修正的CK直接约化方法和Painlevé非标准截断方法的结合不仅能对变系数偏微分方程轻松求解, 还可节省计算时间, 并广泛应用于若干数学物理模型中.