Integrable couplings of vector AKNS soliton equations

2005 ◽  
Vol 46 (3) ◽  
pp. 033507 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Soliton Equations ◽  
Integrable Couplings

SUBALGEBRAS OF THE LIE ALGEBRA R6 AND THEIR APPLICATIONS

2007 ◽  
Vol 21 (22) ◽  
pp. 3809-3824 ◽  
Author(s):  
YU-FENG ZHANG ◽  
EN-GUI FAN
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Systems ◽  
Hamiltonian Structure ◽  
Hamiltonian Structures ◽  
Soliton Theory ◽  
Soliton Equations ◽  
Integrable Couplings ◽  
Potential Component ◽  
Lagrange Mechanics ◽  
Important Significance

As we all know, the Hamiltonian systems are the same describing forms as Newton mechanics and Lagrange mechanics. Therefore, researching for a new Hamiltonian structure of the soliton equations has important significance. In the paper, firstly, with the help of the Lie algebra R6, a few types of subalgebras are constructed, from which the corresponding equivalent tensor systems are given. For their applications, two integrable couplings hierarchies along with the multi-potential component functions generated from the soliton theory and the Virasoro symmetric algebra are obtained. Secondly, the Hamiltonian structures of the above integrable couplings are worked out, which may become another describing expression for the Newton and Lagrange mechanics. In particular, one of the integrable couplings presented above reduces to the famous AKNS hierarchy of soliton equations.

A HIERARCHY OF DISCRETE INTEGRABLE SYSTEM AND ITS COUPLED ENLARGED INTEGRABLE SYSTEM

2007 ◽  
Vol 21 (02n03) ◽  
pp. 155-161
Author(s):  
HAI-YONG DING ◽  
XIANG TIAN ◽  
XI-XIANG XU ◽  
HONG-XIANG YANG
Keyword(s):  
Spectral Problem ◽  
Integrable System ◽  
Soliton Equations ◽  
Discrete Integrable System ◽  
Zero Curvature Representation ◽  
Zero Curvature ◽  
Integrable Lattice ◽  
Integrable Couplings

A hierarchy of nonlinear integrable lattice soliton equations is derived from a discrete spectral problem. The hierarchy is proved to have discrete zero curvature representation. Using an enlarging algebra system [Formula: see text], we construct integrable couplings of the resulting hierarchy.

THE MODIFIED TODA LATTICE IN (2+1)-DIMENSIONS AND INTEGRABLE COUPLING SYSTEMS

2006 ◽  
Vol 20 (23) ◽  
pp. 3341-3355
Author(s):  
HONGXIANG YANG ◽  
XIXIANG XU ◽  
XIUZHEN LI ◽  
CHANGSHENG LI
Keyword(s):  
Toda Lattice ◽  
Soliton Equations ◽  
Spectral Problems ◽  
Higher Dimensional ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Integrable Coupling

By considering a discrete isospectral problem [H.-X. Yang et al., Phys. Lett. A338, 117 (2005)], integrable positive and negative lattice equations are derived, from which the modified (2+1)-dimensional Toda lattice is obtained. The method of enlarging spectral problems to construct the integrable couplings for lattice soliton equations is extended to higher-dimensional systems. Illustrating by examples, the positive and negative integrable couplings of the resulting lattice hierarchy and three classes of integrable couplings of (2+1)-dimensional mToda lattice are discussed.

TWO HIERARCHIES OF NONLINEAR SOLITON EQUATIONS, NEW INTEGRABLE SYMPLECTIC MAP AND DISCRETE INTEGRABLE COUPLINGS

2010 ◽  
Vol 24 (24) ◽  
pp. 4821-4834
Author(s):  
YE-PENG SUN ◽  
HONG-QING ZHAO
Keyword(s):  
Spectral Problem ◽  
Lie Algebra ◽  
Hamiltonian Systems ◽  
Integrable Hamiltonian Systems ◽  
Soliton Equations ◽  
Completely Integrable ◽  
Direct Sum ◽  
Symplectic Map ◽  
Integrable Couplings ◽  
Completely Integrable Hamiltonian Systems

Two hierarchies of nonlinear soliton equations are derived from a discrete spectral problem. It is shown that the hierarchies are completely integrable Hamiltonian systems. Moreover, a new integrable symplectic map is obtained using the binary nonlinearization method. With the help of semi-direct sum of Lie algebra, discrete integrable couplings are constructed.

scholarly journals Tri-Integrable Couplings of the Giachetti-Johnson Soliton Hierarchy as well as Their Hamiltonian Structure

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Lei Wang ◽  
Ya-Ning Tang
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Hamiltonian Structures ◽  
Soliton Equations ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings

Based on zero curvature equations from semidirect sums of Lie algebras, we construct tri-integrable couplings of the Giachetti-Johnson (GJ) hierarchy of soliton equations and establish Hamiltonian structures of the resulting tri-integrable couplings by the variational identity.

A NEW INTEGRABLE LATTICE HIERARCHY AND ITS TWO DISCRETE INTEGRABLE COUPLINGS

2008 ◽  
Vol 22 (24) ◽  
pp. 2411-2419
Author(s):  
LING LI ◽  
HUANHE DONG
Keyword(s):  
Soliton Equations ◽  
Spectral Problems ◽  
Integrable Lattice ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Liouville Integrable

Starting from a discrete isospectral problem, a hierarchy of nonlinear Liouville integrable lattice soliton equations are derived. Through enlarging associated spectral problems, two kinds of discrete integrable couplings are constructed for the resulting lattice hierarchy.

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