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

By Angel Durán, Denys Dutykh and Dimitrios Mitsotakis

Abstract

32 pages, 5 figures, 1 table, 67 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/International audienceThe 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 flavours depending on the goals of the modeller. 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

Topics: long dispersive wave, Boussinesq equations, surface waves, multi-symplectic structure, 76B15 (primary), 76B25 (secondary)47.35.Bb (primary), 47.35.Fg (secondary), [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph], [PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI], [NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], [PHYS.PHYS.PHYS-AO-PH]Physics [physics]/Physics [physics]/Atmospheric and Oceanic Physics [physics.ao-ph]
Publisher: 'Elsevier BV'
Year: 2019
DOI identifier: 10.1016/j.physd.2018.11.007
OAI identifier: oai:HAL:hal-01930748v1
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