Canonical structure of the coupled Korteweg–de Vries equations

2004 ◽  
Vol 82 (6) ◽  
pp. 459-466 ◽  
Author(s):  
B Talukdar ◽  
S Ghosh ◽  
J Shamanna
Keyword(s):  
Inverse Problem ◽  
Lagrangian Density ◽  
Hamiltonian Structure ◽  
Variational Calculus ◽  
Lax Pair ◽  
Theory Of Constraints ◽  
Canonical Structure ◽  
De Vries ◽  
Order Degenerate ◽  
Degenerate Lagrangian

The inverse problem of variational calculus is solved for the coupled Korteweg–de Vries equations resulting from a complex Lax pair. The system is found to be characterized by a second-order degenerate Lagrangian density having some common feature with the well-known Morse–Feshbach Lagrangian. The Hamiltonian structure is examined using Dirac's theory of constraints. PACS Nos.: 47.20.Ky, 42.81.Dp

Generalised (2+1)-dimensional Super MKdV Hierarchy for Integrable Systems in Soliton Theory

2015 ◽  
Vol 5 (3) ◽  
pp. 256-272 ◽  
Author(s):  
Huanhe Dong ◽  
Kun Zhao ◽  
Hongwei Yang ◽  
Yuqing Li
Keyword(s):  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Lie Superalgebra ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Scientific Applications ◽  
Soliton Theory ◽  
De Vries ◽  
Pair Method ◽  
Mkdv Hierarchy

AbstractMuch attention has been given to constructing Lie and Lie superalgebra for integrable systems in soliton theory, which often have significant scientific applications. However, this has mostly been confined to (1+1)-dimensional integrable systems, and there has been very little work on (2+1)-dimensional integrable systems. In this article, we construct a class of generalised Lie superalgebra that differs from more common Lie superalgebra to generate a (2+1)-dimensional super modified Korteweg-de Vries (mKdV) hierarchy, via a generalised Tu scheme based on the Lax pair method where the Hamiltonian structure derives from a generalised supertrace identity. We also obtain some solutions of the (2+1)-dimensional mKdV equation using the G′/G2 method.

scholarly journals The Modified Korteweg-de Vries Hierarchy: Lax Pair Representation and Bi-Hamiltonian Structure

2009 ◽  
Vol 64 (3-4) ◽  
pp. 171-179 ◽  
Author(s):  
Amitava Choudhuri ◽  
Benoy Talukdar ◽  
Umapada Das
Keyword(s):  
Canonical Form ◽  
Poisson Structure ◽  
Hamiltonian Structure ◽  
Lax Pair ◽  
Action Principle ◽  
Miura Transformation ◽  
De Vries ◽  
Lax Pair Representation ◽  
Pair Representation ◽  
Mkdv Hierarchy

Abstract We consider equations in the modified Korteweg-de Vries (mKdV) hierarchy and make use of the Miura transformation to construct expressions for their Lax pair. We derive a Lagrangian-based approach to study the bi-Hamiltonian structure of the mKdV equations. We also show that the complex modified KdV (cmKdV) equation follows from the action principle to have a Lagrangian representation. This representation not only provides a basis to write the cmKdV equation in the canonical form endowed with an appropriate Poisson structure but also help to construct a semianalytical solution of it. The solution obtained by us may serve as a useful guide for purely numerical routines which are currently being used to solve the cmKdV equation

(1+1)-dimensional m-cKdV, g-cKdV integrable systems, and (2+1)-dimensional m-cKdV hierarchy

2008 ◽  
Vol 86 (12) ◽  
pp. 1367-1380 ◽  
Author(s):  
Y Zhang ◽  
H Tam
Keyword(s):  
Integrable Systems ◽  
Linear Functional ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Integrable Model ◽  
Lax Pair ◽  
The Self ◽  
Hamiltonian Structures ◽  
Yang Mills ◽  
Higher Dimensional

A few isospectral problems are introduced by referring to that of the cKdV equation hierarchy, for which two types of integrable systems called the (1 + 1)-dimensional m-cKdV hierarchy and the g-cKdV hierarchy are generated, respectively, whose Hamiltonian structures are also discussed by employing a linear functional and the quadratic-form identity. The corresponding expanding integrable models of the (1 + 1)-dimensional m-cKdV hierarchy and g-cKdV hierarchy are obtained. The Hamiltonian structure of the latter one is given by the variational identity, proposed by Ma Wen-Xiu as well. Finally, we use a Lax pair from the self-dual Yang–Mills equations to deduce a higher dimensional m-cKdV hierarchy of evolution equations and its Hamiltonian structure. Furthermore, its expanding integrable model is produced by the use of a enlarged Lie algebra.PACS Nos.: 02.30, 03.40.K

MULTIPLE LOCAL LAGRANGIANS FOR n-COMPONENT SUPER-KORTEWEG–DE VRIES-TYPE BI-HAMILTONIAN SYSTEMS

2010 ◽  
Vol 25 (05) ◽  
pp. 1069-1078 ◽  
Author(s):  
ÖMER OĞUZ ◽  
DEVRIM YAZICI
Keyword(s):  
Hamiltonian Systems ◽  
Hamiltonian Structure ◽  
Velocity Fields ◽  
Kinetic Term ◽  
Lagrangian Formalism ◽  
Momentum Maps ◽  
Miura Transformation ◽  
De Vries ◽  
Superintegrable Systems

The multiple Lagrangian formalism is constructed for n-component Korteweg–de Vries (KdV) type superintegrable systems. They all admit bi-Hamiltonian structure. The first two Lagrangians are local and degenerate. They contain Clebsch potentials for velocity fields and momentum maps in kinetic term. The first local Lagrangian for n-component supermodified KdV (smKdV) is also obtained by employing the multicomponent super-Miura transformation.

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