Zero curvature representation, bi-Hamiltonian structure, and an integrable hierarchy for the Zakharov-Ito system

2015 ◽  
Vol 56 (6) ◽  
pp. 063503 ◽  
Author(s):  
Mathew Baxter ◽  
S. Roy Choudhury ◽  
Robert A. Van Gorder
Keyword(s):  
Hamiltonian Structure ◽  
Integrable Hierarchy ◽  
Zero Curvature Representation ◽  
Zero Curvature

scholarly journals A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Spectral Problem ◽  
Lie Algebra ◽  
Difference Equations ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Zero Curvature Representation ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Liouville Integrable

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

scholarly journals A Complex Integrable Hierarchy and Its Hamiltonian Structure for Integrable Couplings of WKI Soliton Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Fajun Yu ◽  
Shuo Feng ◽  
Yanyu Zhao
Keyword(s):  
Hamiltonian Structure ◽  
Direct Application ◽  
Integrable Hierarchy ◽  
Soliton Equation ◽  
Zero Curvature ◽  
Coupling System ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Nonlinear Composite ◽  
Integrable Coupling

We generate complex integrable couplings from zero curvature equations associated with matrix spectral problems in this paper. A direct application to the WKI spectral problem leads to a novel soliton equation hierarchy of integrable coupling system; then we consider the Hamiltonian structure of the integrable coupling system. We select theU¯,V¯and generate the nonlinear composite parts, which generate new extended WKI integrable couplings. It is also indicated that the method of block matrix is an efficient and straightforward way to construct the integrable coupling system.

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 A Family of Integrable Different-Difference Equations, Its Hamiltonian Structure, and Darboux-Bäcklund Transformation

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Xi-Xiang Xu ◽  
Meng Xu
Keyword(s):  
Exact Solution ◽  
Difference Equations ◽  
Hamiltonian Structure ◽  
Bäcklund Transformation ◽  
Lax Pair ◽  
Liouville Integrability ◽  
Backlund Transformation ◽  
Zero Curvature Representation ◽  
Zero Curvature ◽  
Matrix Spectral Problem

An integrable family of the different-difference equations is derived from a discrete matrix spectral problem by the discrete zero curvature representation. Hamiltonian structure of obtained integrable family is established. Liouville integrability for the obtained family of discrete Hamiltonian systems is proved. Based on the gauge transformation between the Lax pair, a Darboux-Bäcklund transformation of the first nonlinear different-difference equation in obtained family is deduced. Using this Darboux-Bäcklund transformation, an exact solution is presented.

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.

ZERO CURVATURE CONDITION OF OSp(2/2) AND THE ASSOCIATED SUPERGRAVITY THEORY

1992 ◽  
Vol 07 (18) ◽  
pp. 4293-4311 ◽  
Author(s):  
ASHOK DAS ◽  
WEN-JUI HUANG ◽  
SHIBAJI ROY
Keyword(s):  
Current Algebra ◽  
Hamiltonian Structure ◽  
Operator Product ◽  
Curvature Condition ◽  
Supergravity Theory ◽  
Kdv Equations ◽  
Zero Curvature ◽  
Anomaly Equation ◽  
Graded Lie Algebra ◽  
The One

The N=2 fermionic extensions of the KdV equations are derived from the zero curvature condition associated with the graded Lie algebra of OSp(2/2). These equations lead to two bi-Hamiltonian systems, one of which is supersymmetric. We also derive the one-parameter family of N=2 supersymmetric KdV equations without a bi-Hamiltonian structure in this approach. Following our earlier proposal, we interpret the zero curvature condition as a gauge anomaly equation which brings out the underlying current algebra for the corresponding 2D supergravity theory. This current algebra is then used to obtain the operator product expansions of various fields of this theory.

A generalized integrable hierarchy related to the relativistic Toda lattice: Hamiltonian structure, Darboux transformation, soliton solution and conservation law

2021 ◽  
Author(s):  
Zhiguo Xu
Keyword(s):  
Darboux Transformation ◽  
Hamiltonian Structure ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Integrable Hierarchy ◽  
Integrable Lattice ◽  
Pass Through ◽  
Matrix Spectral Problem ◽  
Relativistic Toda Lattice ◽  
Toda Lattice Hierarchy

Starting from a more generalized discrete [Formula: see text] matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the [Formula: see text]-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete [Formula: see text]-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.

scholarly journals The Strict AKNS Hierarchy: Its Structure and Solutions

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
G. F. Helminck
Keyword(s):  
Commutative Algebra ◽  
Loop Space ◽  
Integrable Hierarchy ◽  
Akns Hierarchy ◽  
Zero Curvature ◽  
Lax Equations

We discuss an integrable hierarchy of compatible Lax equations that is obtained by a wider deformation of a commutative algebra in the loop space ofsl2than that in the AKNS case and whose Lax equations are based on a different decomposition of this loop space. We show the compatibility of these Lax equations and that they are equivalent to a set of zero curvature relations. We present a linearization of the system and conclude by giving a wide construction of solutions of this hierarchy.

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