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

N-SOLITON-LIKE SOLUTIONS AND BÄCKLUND TRANSFORMATIONS FOR A NON-ISOSPECTRAL AND VARIABLE-COEFFICIENT MODIFIED KORTEWEG-DE VRIES EQUATION

2011 ◽  
Vol 25 (05) ◽  
pp. 723-733 ◽  
Author(s):  
QIAN FENG ◽  
YI-TIAN GAO ◽  
XIANG-HUA MENG ◽  
XIN YU ◽  
ZHI-YUAN SUN ◽  
...  
Keyword(s):  
Variable Coefficient ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Wronskian Technique ◽  
Backlund Transformation ◽  
Bilinear Bäcklund Transformation ◽  
De Vries ◽  
Bilinear Equations

A non-isospectral and variable-coefficient modified Korteweg–de Vries (mKdV) equation is investigated in this paper. Starting from the Ablowitz–Kaup–Newell–Segur procedure, the Lax pair is established and the Bäcklund transformation in original variables is also derived. By a dependent variable transformation, the non-isospectral and variable-coefficient mKdV equation is transformed into bilinear equations, by virtue of which the N-soliton-like solution is obtained. In addition, the bilinear Bäcklund transformation gives a one-soliton-like solution from a vacuum one. Furthermore, the N-soliton-like solution in the Wronskian form is constructed and verified via the Wronskian technique.

scholarly journals Geometric Poisson brackets on Grassmannians and conformal spheres

2012 ◽  
Vol 142 (3) ◽  
pp. 525-561 ◽  
Author(s):  
M. Eastwood ◽  
G. Marí Beffa
Keyword(s):  
Poisson Bracket ◽  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Coupled System ◽  
Differential Invariants ◽  
Poisson Brackets ◽  
Moving Frames ◽  
Kdv Equations ◽  
De Vries ◽  
Natural Frame

We relate the geometric Poisson brackets on the 2-Grassmannian in ℝ4 and on the (2, 2) Möbius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Möbius sphere does not restrict to the space of differential invariants of Schwarzian type. But when the concept of conformal natural frame is transported from the conformal sphere into the Grassmannian, and the Poisson bracket is written in terms of the Grassmannian natural frame, it restricts and results in either a decoupled system or a complexly coupled system of Korteweg–de Vries (KdV) equations, depending on the character of the invariants. We also show that the bi-Hamiltonian Grassmannian geometric brackets are equivalent to the non-commutative KdV bi-Hamiltonian structure. Both integrable systems and Hamiltonian structure can be brought back to the conformal sphere.

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

Two synthetical five-component nonlinear integrable systems: Darboux transformations and applications

2020 ◽  
Vol 34 (32) ◽  
pp. 2050314
Author(s):  
Xin Chen ◽  
Qi-Lao Zha
Keyword(s):  
Diffusion Equation ◽  
Integrable Systems ◽  
Smooth Function ◽  
Reaction Diffusion ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Reaction Diffusion Equation ◽  
Darboux Transformations ◽  
Arbitrary Smooth Function ◽  
Matrix Spectral Problem

A generalized [Formula: see text] matrix spectral problem is investigated to generate two five-component nonlinear integrable systems, which involve an arbitrary smooth function. These systems are proven integrable in the sense of Lax pair. As the reduction cases, a four-component reaction diffusion equation and a four-component modified Korteweg-de Vries (mKdV) equation are solved by Darboux transformation approach.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

Multiple Soliton Solutions for a Variety of Coupled Modified Korteweg–de Vries Equations

2011 ◽  
Vol 66 (10-11) ◽  
pp. 625-631
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Mkdv Equation ◽  
Resonance Phenomenon ◽  
Hirota’S Bilinear Method ◽  
Bilinear Method ◽  
Multiple Soliton Solutions ◽  
De Vries ◽  
Coupled Equation ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

We make use of Hirota’s bilinear method with computer symbolic computation to study a variety of coupled modified Korteweg-de Vries (mKdV) equations. Multiple soliton solutions and multiple singular soliton solutions are obtained for each coupled equation. The resonance phenomenon of each coupled mKdV equation is proved not to exist.

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