广义的WBKL方程和HS-KdV方程的微分不变量、微分不变方程

Differential Invariants, Differential Invariant Equations for the Generalized WBKL Equation and HS-KdV Equation

以数学物理等领域中两个重要的非线性发展方程为研究对象,即广义的Whitham-Broer-Kaup-Like(WBKL)方程组和广义的HS-KdV方程组.WBKL方程描述了长波在浅水中的双向传播,此方程也可约化为变形的Boussinesq方程、色散长波方程、Whitham-Broer-Kaup方程等.由于WBKL方程和广义的Hirota-Satsuma耦合KdV方程的非线性和经典活动标架法的局限性,运用最新的等变活动标架理论,通过选择合适的群轨道横截面进行规范化,进而得到活动标架,同时借助符号计算系统Maple避免了复杂的高阶微分计算,切实有效地求得了WBKL方程组和广义的HS-KdV方程组的微分不变量、微分不变量代数以及微分不变方程.所得到的结果可用于深入研究WBKL方程和广义的HS-KdV方程解的不变性、等价性和对称性,以及海洋、大气、水波等非线性运动的趋势和规律.

The research object of this paper are two important nonlinear development equations in mathematical physics and other fields,the generalized Whitham-Broer-Kaup-Like(WBKL)equation and the generalized HS-KdV equation.The WBKL equation describes the two-way transposition of long waves in shallow water.This equation can also be reduced to the variant Boussinesq equation,dispersive long-wave equation,Whitham-Broer-Kaup equation,etc.Due to the nonlinearity of the WBKL equation and the generalized HS-KdV equation and the limitation of the classical moving frame method,the latest equivariant moving frames theory is applied to obtain the moving frames by selecting the appropriate group orbital cross section for normalization,while avoiding the complex higher-order differential calculations with the help of the symbolic computing system Maple,which can efficiently find the differential invariants,differential invariant algebra and differential invariant equations of the WBKL equation system,the generalized HS-KdV equation system are obtained.The obtained results can be used to study in depth the invariance,equivalence and symmetry of the solutions of the WBKL equations and the generalized Hirota-Satsuma coupled KdV equations,as well as the trends and laws of nonlinear motions of the ocean,atmosphere and water waves.