Abstract
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.
Highlights
The soliton solutions and their dynamics of lots of nonlinear equations were accurately captured by the variational approximation method [19,20,21,22,23,24,25], which always substitutes some ansatzes into the obtained Lagrange functional, and find the variational parameters by solving the corresponding Euler-Lagrange equations
Because variational principles are the theoretical bases for many kinds of variational methods, it is very important but difficult to seek explicit variational formulations for nonlinear and complex models usually expressed by the nonlinear Partial differential equations (PDEs)
In the second and third parts, different groups of variational principles have been successfully constructed for the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively, by the calculus of variations and designing skillfully trial-Lagrange functionals
Summary
Partial differential equations (PDEs) are usually used to model different phenomena in nonlinear sciences, ranging from physics to mechanics, biology, chemistry, meteorology, ocean, and so on [1,2,3]. When contrasted with other numerical or analytical methods, variational-based methods show a lot of advantages They can be used in investigating practical problems from a global perspective and provide physical insight into the nature of the solutions. They can help to study the symmetries and conserved quantities for the discussed nonlinear problems. The obtained solutions are the best among all possible trial-functions and require much less strong local differentiability of variables than the methods that directly solve PDEs. Because variational principles are the theoretical bases for many kinds of variational methods, it is very important but difficult to seek explicit variational formulations for nonlinear and complex models usually expressed by the nonlinear PDEs. It is an inverse problem to directly find variational principles from a set of known equations by calculus rules. Finding variational principles for them is of great value and maybe helpful for harbor and coastal designs
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