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

2020 ◽  
Author(s):  
A Durán ◽  
D Dutykh ◽  
Dimitrios Mitsotakis
Keyword(s):  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Boussinesq Equations ◽  
Type Systems ◽  
Symmetry Groups ◽  
Conserved Quantities ◽  
Mathematical Properties ◽  
Different Types ◽  
Xxist Century ◽  
Boussinesq Models

© 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.

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

2020 ◽  
Author(s):  
A Durán ◽  
D Dutykh ◽  
Dimitrios Mitsotakis
Keyword(s):  
Symplectic Structure ◽  
Hamiltonian Structure ◽  
Boussinesq Equations ◽  
Type Systems ◽  
Symmetry Groups ◽  
Conserved Quantities ◽  
Mathematical Properties ◽  
Different Types ◽  
Xxist Century ◽  
Boussinesq Models

© 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.

scholarly journals Integrability in Nonlinear Realization Scheme

1997 ◽  
Vol 12 (01) ◽  
pp. 231-236 ◽  
Author(s):  
R. P. Malik
Keyword(s):  
Hamiltonian Structure ◽  
Geometric Approach ◽  
Boussinesq Equations ◽  
Lax Pair ◽  
Conserved Quantities ◽  
Nonlinear Realization ◽  
Zero Curvature Representation ◽  
Zero Curvature ◽  
Key Features

In the framework of universal geometric approach of nonlinear realization method, some of the key features of the integrability properties of the Boussinesq equations, connected with the W3 algebra of Zamolodchikov, are discussed. The geometrical origins for these equations, its second Hamiltonian structure, Lax-pair formulation, zero-curvature representation, involving conserved quantities, etc., have also been concisely dealt with under the nonlinear realization scheme.

scholarly journals On the multi-symplectic structure of Boussinesq-type systems. II: Geometric discretization

2020 ◽  
Author(s):  
A Durán ◽  
D Dutykh ◽  
Dimitrios Mitsotakis
Keyword(s):  
Symplectic Structure ◽  
Numerical Approximation ◽  
Method Of Lines ◽  
Type Systems ◽  
Well Posedness ◽  
Solitary Wave Solutions ◽  
Model Surface ◽  
Hamiltonian Structures ◽  
Surface Wave Propagation ◽  
The Method Of Lines

© 2019 Elsevier B.V. In this paper we consider the numerical approximation of systems of BOUSSINESQ-type to model surface wave propagation. Some theoretical properties of these systems (multi-symplectic and HAMILTONIAN formulations, well-posedness and existence of solitary-wave solutions)were previously analysed by the authors in Part I. As a second part of the study, considered here is the construction of geometric schemes for the numerical integration. By using the method of lines, the geometric properties, based on the multi-symplectic and HAMILTONIAN structures, of different strategies for the spatial and time discretizations are discussed and illustrated.

A FRACTIONAL KdV HIERARCHY

1991 ◽  
Vol 06 (17) ◽  
pp. 1561-1573 ◽  
Author(s):  
IOANNIS BAKAS ◽  
DIDIER A. DEPIREUX
Keyword(s):  
Hamiltonian Structure ◽  
Boussinesq Equation ◽  
Operator Algebras ◽  
Nonlinear Differential Equations ◽  
Conserved Quantities ◽  
Gauge Groups ◽  
Yang Mills ◽  
Kdv Hierarchy ◽  
Alternative Set ◽  
New System

We construct a new system of integrable nonlinear differential equations associated with. the operator algebra [Formula: see text] of Polyakov. Its members are fractional generalizations of KdV type flows corresponding to an alternative set of constraints on the 2-dim. SL(3) gauge connections. We obtain the first non-trivial examples by dimensional reductiion from self-dual Yang–Mills and then generate recursively the entire hierarchy and its conserved quantities using a bi-Hamiltonian structure. Certain relations with the Boussinesq equation are also discussed together with possible generalizations of the formalism to SL (N) gauge groups and [Formula: see text] operator algebras with arbitrary N and l.

Experimental Investigation of Spatial Pattern Formation in Physical Systems of Activator Inhibitor Type

1988 ◽  
Vol 43 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Hans-Georg Purwins ◽  
Christian Radehaus ◽  
Jürgen Berkemeier
Keyword(s):  
Reaction Diffusion ◽  
Type Systems ◽  
Reaction Diffusion Equation ◽  
Electrical Networks ◽  
Stable States ◽  
Mathematical Properties ◽  
Dimensional Network ◽  
Current Voltage ◽  
Inhibitor Type ◽  
Activator Inhibitor Type

Abstract We investigate experimentally stationary stable states of activator (w) inhibitor (υ) type systems corresponding to the reaction diffusion equation δ · υ̇ = Δυ + w - υ; ẇ = σ Δw + f(w) - υ; δ, σ = const > 0 with f(w) monotonically increasing for small and decreasing for large |w|. We first describe some general mathematical properties and present qualitative results obtained from numerical calculations. We then investigate experimentally electrical networks described by the spatially discretized version of the above equation. Calculation and experiment are in good agreement. We also interprete a two dimensional-network as an equivalent circuit for a composite material consisting of a linear and a nonlinear layer with an s-shaped current density electric field characteristic. This model is used for a phenomenological description of spatial structures and global current voltage characteristics observed experimentally in pin-diode like and gas discharge devices. The model accounts very well for the experimental results obtained so far. It is concluded that the above equation and the corresponding experimental setup are of great interest for fundamental investigations of self con­trolled processes in nature.

scholarly journals Symplectic connections and the linearisation of Hamiltonian systems

1991 ◽  
Vol 117 (3-4) ◽  
pp. 329-380 ◽  
Author(s):  
J. E. Marsden ◽  
T. Ratiu ◽  
G. Raugel
Keyword(s):  
Hamiltonian System ◽  
Symplectic Structure ◽  
Equilibrium Solution ◽  
Hamiltonian Structure ◽  
Second Variation ◽  
First Variation ◽  
Space Construction ◽  
Symplectic Connections ◽  
The First Variation ◽  
The Second Variation

SynopsisThis paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie–Poisson systems in particular.

scholarly journals N = 2 SUPER W3 ALGEBRA AND N = 2 SUPER BOUSSINESQ EQUATIONS

1995 ◽  
Vol 10 (02) ◽  
pp. 253-288 ◽  
Author(s):  
E. IVANOV ◽  
S. KRIVONOS ◽  
R.P. MALIK
Keyword(s):  
Hamiltonian Structure ◽  
Boussinesq Equation ◽  
Boussinesq Equations ◽  
Geometric Interpretation ◽  
Nonlinear Realization ◽  
Linear Formula ◽  
Reduction Approach

We study classical N=2 super W3 algebra and its interplay with N=2 supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs-covariant reduction approach. These techniques have been previously used by us in the bosonic W3 case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general N=2 super Boussinesq equation and two kinds of the modified N=2 super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to certain coset manifolds of linear [Formula: see text] symmetry associated with N=2 super W3. We discuss the integrability properties of the equations obtained and their correspondence with the formulation based on the notion of the second Hamiltonian structure.

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