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Open AccessArticle

Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

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College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410073, China
2
College of Computer, National University of Defense Technology, Changsha 410073, China
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Author to whom correspondence should be addressed.
Symmetry 2020, 12(5), 850; https://doi.org/10.3390/sym12050850
Received: 4 April 2020 / Revised: 7 May 2020 / Accepted: 8 May 2020 / Published: 22 May 2020
It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of variational principles are successfully constructed for two kinds of coupled nonlinear equations in shallow water, i.e., the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively. Both of them contain many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Subsequently, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations. The established variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities for the equations considered, and might find lots of applications in numerical simulation. View Full-Text
Keywords: variational principle; calculus of variations; Broer-Kaup equations; (2+1)-dimensional dispersive long-wave equations variational principle; calculus of variations; Broer-Kaup equations; (2+1)-dimensional dispersive long-wave equations
MDPI and ACS Style

Cao, X.-Q.; Guo, Y.-N.; Hou, S.-C.; Zhang, C.-Z.; Peng, K.-C. Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water. Symmetry 2020, 12, 850.

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