Nonlinear Self-Adjointness, Conservation Laws and Soliton-Cnoidal Wave Interaction Solutions of (2+1)-Dimensional Modified Dispersive Water-Wave System

2017 ◽  
Vol 67 (1) ◽  
pp. 15 ◽  
Author(s):  
Ya-Rong Xia ◽  
Xiang-Peng Xin ◽  
Shun-Li Zhang
Keyword(s):  
Conservation Laws ◽  
Water Wave ◽  
Wave Interaction ◽  
Wave System ◽  
Cnoidal Wave ◽  
Interaction Solutions

scholarly journals Some invariant solutions and conservation laws of a type of long-water wave system

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiangzhi Zhang ◽  
Yufeng Zhang
Keyword(s):  
Conservation Laws ◽  
Water Wave ◽  
Lax Pair ◽  
Wave System ◽  
Series Solutions ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Generalized System ◽  
Group Invariant Solutions ◽  
Similarity Reductions

AbstractWe propose a generalized long-water wave system that reduces to the standard water wave system. We also obtain the Lax pair and symmetries of the generalized shallow-water wave system and single out some their similarity reductions, group-invariant solutions, and series solutions. We further investigate the corresponding self-adjointness and the conservation laws of the generalized system.

Residual Symmetry and Explicit Soliton–Cnoidal Wave Interaction Solutions of the (2+1)-Dimensional KdV–mKdV Equation

2016 ◽  
Vol 71 (4) ◽  
pp. 351-356 ◽  
Author(s):  
Wenguang Cheng ◽  
Biao Li
Keyword(s):  
Elliptic Integral ◽  
Wave Interaction ◽  
Elliptic Functions ◽  
Mkdv Equation ◽  
Jacobi Elliptic Functions ◽  
Cnoidal Wave ◽  
Residual Symmetry ◽  
The Third ◽  
Interaction Solutions ◽  
Consistent Riccati Expansion

AbstractThe truncated Painlevé method is developed to obtain the nonlocal residual symmetry and the Bäcklund transformation for the (2+1)-dimensional KdV–mKdV equation. The residual symmetry is localised after embedding the (2+1)-dimensional KdV–mKdV equation to an enlarged one. The symmetry group transformation of the enlarged system is computed. Furthermore, the (2+1)-dimensional KdV–mKdV equation is proved to be consistent Riccati expansion (CRE) solvable. The soliton–cnoidal wave interaction solution in terms of the Jacobi elliptic functions and the third type of incomplete elliptic integral is obtained by using the consistent tanh expansion (CTE) method, which is a special form of CRE.

Global well-posedness and conservation laws for the water wave interaction equation

1997 ◽  
Vol 127 (2) ◽  
pp. 369-384 ◽  
Author(s):  
Takayoshi Ogawa
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Water Wave ◽  
Wave Interaction ◽  
Energy Functional ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Interaction Equation ◽  
Image Position ◽  
Well Posed

Interaction equations of long and short water wave are considered. It is shown that the Cauchy problem foris locally well posed in the largest space where the three conservationscan be justified. Here E(u,v) is the energy functional associated to the system. By these conservation laws, we establish the global well-posedness of the system in the largest class of initial data.

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