辅助函数法解扩展的BBM方程

Auxiliary Function Method for Solving Extended BBM Equations

摘要: 扩展的BBM方程是一个含有非线性项的偏微分方程。在物理学中，用非线性偏微分方程来描述物理模型非常普遍；在数学中，非线性偏微分方程可以用来证明Poincaré猜想和Calabi猜想的合理性。利用辅助方程，通过行波变换转化为常微分方程后，借助辅助方程来求解转化后的常微分方程，进而可以得到偏微分方程的精确解。为此，通过行波变换及辅助方程的求解思路对BBM方程进行了研究，并得到了该方程双曲正切函数及三角函数形式的精确解。据此可推广应用至其他类似的非线性偏微分方程中。

Abstract: Extended BBM Equations are partial differential equations with nonlinear terms. In physics, it is very common to use nonlinear partial differential equation to describe physical models; in mathe-matics, nonlinear partial differential equation can be used to prove the rationality of Poincar é con-jecture and Calabi conjecture. The auxiliary equation can be transformed into Ordinary differential equation by traveling wave transformation, and then the transformed Ordinary differential equa-tion can be solved by the auxiliary equation, and then the exact solution of partial differential equa-tion can be obtained. For this reason, the BBM equation is studied through traveling wave trans-formation and the idea of solving auxiliary equations, and the exact solutions of the equation in the form of hyperbolic tangent function and Trigonometric functions are obtained. Therefore, it can be extended to other similar nonlinear partial differential equation.