Different Groups of Variational Principles for Whitham-Broer- Kaup Equations in Shallow Water

Because the variational theory is the theoretical basis for many kinds of analytical or numerical methods, it is an essential but difficult task to seek explicit functional formulations whose extrema are sought by the nonlinear and complex models. By the semi-inverse method and designing trial-Lagrange functional skillfully, two different groups of variational principles are constructed for the Whitham-Broer-Kaup equations, which can model a lot of nonlinear shallow-water waves. Furthermore, by a combination of different variational formulations, new families of variational principles are established. The obtained variational principles provide conservation laws in an energy form and are proved correct by minimizing the functionals with the calculus of variations. All variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities, and might find lots of applications in numerical simulation. The procedure reveals that the semi-inverse method is highly efficient and powerful, and can be extended to more other nonlinear equations.