scholarly journals Two Kinds of New Integrable Couplings of the Negative-Order Korteweg-de Vries Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Binlu Feng ◽  
Yufeng Zhang
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Vries Equation ◽  
Kdv Hierarchy ◽  
Kdv Equations ◽  
Loop Algebras ◽  
Negative Order ◽  
Integrable Couplings ◽  
De Vries ◽  
Integrable Coupling

Based on some known loop algebras with finite dimensions, two different negative-order integrable couplings of the negative-order Korteweg-de Vries (KdV) hierarchy of evolution equations are generated by making use of the Tu scheme, from which the corresponding negative-order integrable couplings of the negative-order KdV equations are followed to be obtained. The resulting Hamiltonian structure of one negative integrable coupling is derived from the variational identity.

scholarly journals A New Approach for Generating the TX Hierarchy as well as Its Integrable Couplings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Guangming Wang
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
New Approach ◽  
Zero Curvature ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Integrable Coupling

Tu Guizhang and Xu Baozhi once introduced an isospectral problem by a loop algebra with degree beingλ, for which an integrable hierarchy of evolution equations (called the TX hierarchy) was derived under the frame of zero curvature equations. In the paper, we present a loop algebra whose degrees are2λand2λ+1to simply represent the above isospectral matrix and easily derive the TX hierarchy. Specially, through enlarging the loop algebra with 3 dimensions to 6 dimensions, we generate a new integrable coupling of the TX hierarchy and its corresponding Hamiltonian structure.

scholarly journals Nonlocal Symmetry and Bäcklund Transformation of a Negative-Order Korteweg–de Vries Equation

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Jinxi Fei ◽  
Weiping Cao ◽  
Zhengyi Ma
Keyword(s):  
Vries Equation ◽  
Bäcklund Transformation ◽  
Backlund Transformation ◽  
Nonlocal Symmetry ◽  
Linear Superposition ◽  
Residual Symmetry ◽  
Negative Order ◽  
De Vries ◽  
Finite Transformation ◽  
New Variables

The residual symmetry of a negative-order Korteweg–de Vries (nKdV) equation is derived through its Lax pair. Such residual symmetry can be localized, and the original nKdV equation is extended into an enlarged system by introducing four new variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Furthermore, we localize the linear superposition of multiple residual symmetries and construct n-th Bäcklund transformation for this nKdV equation in the form of the determinants.

Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950054
Author(s):  
H. Wajahat A. Riaz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Order Effects ◽  
Darboux Transformations ◽  
De Vries ◽  
Higher Order Effects

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

The Multi-Component Jaulent-Miodek Hierarchy and its Multi-Component Integrable Coupling System with Two Arbitrary Functions

2012 ◽  
Vol 442 ◽  
pp. 124-128
Author(s):  
Jian Ya Ge ◽  
Tie Cheng Xia
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Loop Algebra ◽  
Simple Loop ◽  
Coupling System ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Integrable Coupling

We devise a new simple loop algebra GM and an isospectral problem. By making use of Tu scheme, the multi-component Jaulent-Miodek (JM) hierarchy is obtained. Furthermore, an expanding loop algebra FM of the loop algebra GM is presented. Based on FM the multi-component integrable couplings system with two arbitrary functions of the multi-component Jaulent-Miodek (JM) hierarchy are worked out. The method can be applied to other nonlinear evolution equations hierarchies.

NONCOMMUTATIVE KdV HIERARCHY

2007 ◽  
Vol 19 (07) ◽  
pp. 677-724 ◽  
Author(s):  
FRANÇOIS TREVES
Keyword(s):  
Evolution Equations ◽  
Vries Equation ◽  
Kdv Equation ◽  
Completely Integrable Systems ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Kdv Hierarchy ◽  
Noncommutative Version ◽  
Constants Of Motion ◽  
Hamiltonian Evolution

The noncommutative version of the Korteweg–de Vries equation studied here is shown to admit infinitely many constants of motion and to give rise to a hierarchy of higher-order Hamiltonian evolution equations, each one the noncommutative version of the commutative KdV equation of the same order. The noncommutative KdV polynomials span, topologically, a maximal Abelian subalgebra of the Lie algebra of noncommutative Bäcklund transformations. Two classes of examples of "completely integrable" systems of evolution equations to which the theory applies are described in the last two sections.

scholarly journals Multi-wave solutions for a non-isospectral KDV-type equation with variable coefficients

2012 ◽  
Vol 16 (5) ◽  
pp. 1476-1479 ◽  
Author(s):  
Sheng Zhang ◽  
Qun Gao ◽  
Qian-An Zong ◽  
Dong Liu
Keyword(s):  
Variable Coefficient ◽  
Function Method ◽  
Evolution Equations ◽  
Vries Equation ◽  
Type Equation ◽  
Variable Coefficients ◽  
Linear Evolution ◽  
Wave Solutions ◽  
De Vries ◽  
Linear Evolution Equations

As a typical mathematical model in fluids and plasmas, Korteweg-de Vries equation is famous. In this paper, the Exp-function method is extended to a nonisos-pectral Korteweg-de Vries type equation with three variable coefficients, and multi-wave solutions are obtained. It is shown that the Expfunction method combined with appropriate ansatz may provide with a straightforward, effective and alternative method for constructing multi-wave solutions of variable-coefficient non-linear evolution equations.

A SOLITON HIERARCHY FROM THE LEVI SPECTRAL PROBLEM AND ITS TWO INTEGRABLE COUPLINGS, HAMILTONIAN STRUCTURE

2009 ◽  
Vol 23 (05) ◽  
pp. 731-739
Author(s):  
YONGQING ZHANG ◽  
YAN LI
Keyword(s):  
Spectral Problem ◽  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Soliton Equation ◽  
Curvature Equation ◽  
Loop Algebras ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Time Part

A soliton-equation hierarchy from the D. Levi spectral problem is obtained under the framework of zero curvature equation. By employing two various multi-component Lie algebras and the loop algebras, we enlarge the Levi spectral problem and the corresponding time-part isospectral problems so that two different integrable couplings are produced. Using the quadratic-form identity yields the Hamiltonian structure of one of the two integrable couplings.

INTEGRABLE COUPLING SYSTEMS OF HAMILTONIAN LATTICE EQUATIONS BY SEMI-DIRECT SUMS OF LIE ALGEBRAS

2010 ◽  
Vol 24 (07) ◽  
pp. 681-694
Author(s):  
LI-LI ZHU ◽  
JUN DU ◽  
XIAO-YAN MA ◽  
SHENG-JU SANG
Keyword(s):  
Eigenvalue Problem ◽  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Liouville Integrability ◽  
Direct Sums ◽  
Direct Sum ◽  
Hamiltonian Lattice ◽  
Integrable Couplings ◽  
Type Lattice ◽  
Integrable Coupling

By considering a discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations are derived. The relation to the Toda type lattice is achieved by variable transformation. With the help of Tu scheme, the Hamiltonian structure of the resulting lattice hierarchy is constructed. The Liouville integrability is then demonstrated. Semi-direct sum of Lie algebras is proposed to construct discrete integrable couplings. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

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