On the multi-symplectic structure of Boussinesq-type systems. I: Derivation and mathematical properties

© 2018 Elsevier B.V. The BOUSSINESQ equations are known since the end of the XIXst century. However, the proliferation of various BOUSSINESQ-type systems started only in the second half of the XXst century. Today they come under various flavors depending on the goals of the modeler. At the beginning of the XXIst century an effort to classify such systems, at least for even bottoms, was undertaken and developed according to both different physical regimes and mathematical properties, with special emphasis, in this last sense, on the existence of symmetry groups and their connection to conserved quantities. Of particular interest are those systems admitting a symplectic structure, with the subsequent preservation of the total energy represented by the HAMILTONIAN. In the present paper a family of BOUSSINESQ-type systems with multi-symplectic structure is introduced. Some properties of the new systems are analyzed: their relation with already known BOUSSINESQ models, the identification of those systems with additional HAMILTONIAN structure as well as other mathematical features like well-posedness and existence of different types of solitary-wave solutions. The consistency of multi-symplectic systems with the full EULER equations is also discussed.

On the multi-symplectic structure of Boussinesq-type systems. I: Derivation and mathematical properties

© 2018 Elsevier B.V. The BOUSSINESQ equations are known since the end of the XIXst century. However, the proliferation of various BOUSSINESQ-type systems started only in the second half of the XXst century. Today they come under various flavors depending on the goals of the modeler. At the beginning of the XXIst century an effort to classify such systems, at least for even bottoms, was undertaken and developed according to both different physical regimes and mathematical properties, with special emphasis, in this last sense, on the existence of symmetry groups and their connection to conserved quantities. Of particular interest are those systems admitting a symplectic structure, with the subsequent preservation of the total energy represented by the HAMILTONIAN. In the present paper a family of BOUSSINESQ-type systems with multi-symplectic structure is introduced. Some properties of the new systems are analyzed: their relation with already known BOUSSINESQ models, the identification of those systems with additional HAMILTONIAN structure as well as other mathematical features like well-posedness and existence of different types of solitary-wave solutions. The consistency of multi-symplectic systems with the full EULER equations is also discussed.

On Some Initial and Initial Boundary Value Problems for Linear and Nonlinear Boussinesq Models

The main concern of this paper is to apply the modified double Laplace decomposition method to a singular generalized modified linear Boussinesq equation and to a singular nonlinear Boussinesq equation. An a priori estimate for the solution is also derived. Some examples are given to validate and illustrate the method.