Differential Invariants, Differential Invariant Equations for the Generalized WBKL Equation and HS-KdV Equation
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 mov-ing 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 in-variant 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.
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