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

Autor: Dimitrios Mitsotakis, Denys Dutykh, Angel Durán
Přispěvatelé: ETSI Telecomunicacion [Valladolid] (ETSI), Université de Valladolid, Universidad de Valladolid [Valladolid] (UVa), Laboratoire de Mathématiques (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Victoria University of Wellington
Rok vydání: 2019
Předmět:
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn]
multi-symplectic structure
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
Mathematical properties
FOS: Physical sciences
Pattern Formation and Solitons (nlin.PS)
76B15 (primary)
76B25 (secondary)47.35.Bb (primary)
47.35.Fg (secondary)
Symmetry group
01 natural sciences
010305 fluids & plasmas
symbols.namesake
Mathematics - Analysis of PDEs
[NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS]
Hamiltonian structure
0103 physical sciences
FOS: Mathematics
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
[NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI]
[PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph]
Total energy
Boussinesq equations
010306 general physics
Mathematical Physics
Mathematics
[PHYS.PHYS.PHYS-AO-PH]Physics [physics]/Physics [physics]/Atmospheric and Oceanic Physics [physics.ao-ph]
Nonlinear Sciences - Exactly Solvable and Integrable Systems
Fluid Dynamics (physics.flu-dyn)
long dispersive wave
Statistical and Nonlinear Physics
Mathematical Physics (math-ph)
Physics - Fluid Dynamics
16. Peace & justice
Condensed Matter Physics
Nonlinear Sciences - Pattern Formation and Solitons
surface waves
Conserved quantity
Euler equations
Algebra
symbols
Exactly Solvable and Integrable Systems (nlin.SI)
Hamiltonian (quantum mechanics)
Analysis of PDEs (math.AP)
Symplectic geometry
Zdroj: Physica D: Nonlinear Phenomena
Physica D: Nonlinear Phenomena, Elsevier, 2019, 388, pp.10-21. ⟨10.1016/j.physd.2018.11.007⟩
ISSN: 0167-2789
DOI: 10.1016/j.physd.2018.11.007
Popis: The Boussinesq equations are known since the end of the XIXst century. However, the proliferation of various \textsc{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.
32 pages, 5 figures, 1 table, 67 references. Other author's papers can be downloaded at http://www.denys-dutykh.com/
Databáze: OpenAIRE
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