scholarly journals Self-Consistent Sources and Conservation Laws for Nonlinear Integrable Couplings of the Li Soliton Hierarchy

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Han-yu Wei ◽  
Tie-cheng Xia
Keyword(s):  
Conservation Laws ◽  
Lie Algebras ◽  
Soliton Hierarchy ◽  
Self Consistent ◽  
Integrable Couplings ◽  
Integrable Coupling

New explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Li soliton hierarchy are obtained. Then, the nonlinear integrable couplings of Li soliton hierarchy with self-consistent sources are established. Finally, we present the infinitely many conservation laws for the nonlinear integrable coupling of Li soliton hierarchy.

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.

scholarly journals Two Expanding Integrable Models of the Geng-Cao Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xiurong Guo ◽  
Yufeng Zhang ◽  
Xuping Zhang
Keyword(s):  
Heat Equation ◽  
Lie Algebras ◽  
Burgers Equation ◽  
Integrable Model ◽  
Integrable Models ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Generalized Burgers Equation ◽  
Integrable Couplings ◽  
Integrable Coupling

As far as linear integrable couplings are concerned, one has obtained some rich and interesting results. In the paper, we will deduce two kinds of expanding integrable models of the Geng-Cao (GC) hierarchy by constructing different 6-dimensional Lie algebras. One expanding integrable model (actually, it is a nonlinear integrable coupling) reduces to a generalized Burgers equation and further reduces to the heat equation whose expanding nonlinear integrable model is generated. Another one is an expanding integrable model which is different from the first one. Finally, the Hamiltonian structures of the two expanding integrable models are obtained by employing the variational identity and the trace identity, respectively.

KN EQUATIONS HIERARCHY WITH SELF-CONSISTENT SOURCES AND ITS INTEGRABLE COUPLINGS

2011 ◽  
Vol 25 (25) ◽  
pp. 3325-3335
Author(s):  
FA-JUN YU ◽  
JIN-CAI CHANG
Keyword(s):  
Lie Algebra ◽  
Kronecker Product ◽  
Loop Algebra ◽  
Soliton Hierarchy ◽  
Self Consistent ◽  
Integrable Couplings

A hierarchy of the KN equations with self-consistent sources is derived with the Lie algebra sl(4). As an application example, the integrable couplings of the KN soliton hierarchy with self-consistent sources are constructed by using of Kronecker product and loop algebra [Formula: see text].

scholarly journals A Few Integrable Couplings of Some Integrable Systems and (2+1)-Dimensional Integrable Hierarchies

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Binlu Feng ◽  
Yufeng Zhang ◽  
Huanhe Dong
Keyword(s):  
Lie Algebras ◽  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Coupling Model ◽  
High Dimensional ◽  
Akns Hierarchy ◽  
Integrable Couplings ◽  
Integrable Coupling

Two high-dimensional Lie algebras are presented for which four (1+1)-dimensional expanding integrable couplings of the D-AKNS hierarchy are obtained by using the Tu scheme; one of them is a united integrable coupling model of the D-AKNS hierarchy and the AKNS hierarchy. Then (2+1)-dimensional DS hierarchy is derived by using the TAH scheme; in particular, the integrable couplings of the DS hierarchy are obtained.

THE ALGEBRAIC STRUCTURE OF ZERO CURVATURE REPRESENTATIONS ASSOCIATED WITH INTEGRABLE COUPLINGS

2008 ◽  
Vol 23 (09) ◽  
pp. 1309-1325 ◽  
Author(s):  
LIN LUO ◽  
WEN-XIU MA ◽  
EN-GUI FAN
Keyword(s):  
Lie Algebras ◽  
Algebraic Structure ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Commutation Relations ◽  
Zero Curvature ◽  
Integrable Couplings ◽  
Integrable Coupling

The commutator of enlarged vector fields was explicitly computed for integrable coupling systems associated with semidirect sums of Lie algebras. An algebraic structure of zero curvature representations is then established for such integrable coupling systems. As an application example of this algebraic structure, the commutation relations of Lax operators corresponding to the enlarged isospectral and nonisospectral AKNS flows are worked out, and thus a τ-symmetry algebra for the AKNS integrable couplings is engendered from this theory.

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.

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.

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