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

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.

Abundant Soliton Structures to the ( 2 + 1 )-Dimensional Heisenberg Ferromagnetic Spin Chain Dynamical Model

“…The VM. Aided by the semi-inverse method [39][40][41][42][43][44][45], the variational principle of Eq. ( 4) can be developed as…”

Study on the explicit solutions of the Benney–Luke equation via the variational direct method

“…Step 3: So the variational principle of Equation (2.3) can be developed via the semi-inverse method, [11][12][13][14][15][16][17][18][19][20][21][22][23] which reads as follows:…”

“… So the variational principle of Equation () can be developed via the semi‐inverse method, 11–23 which reads as follows: …”