Hamiltonian structures of nonlinear evolution equations connected with a polynomial pencil

1986 ◽  
Vol 34 (5) ◽  
pp. 1923-1932 ◽  
Author(s):  
I. T. Gadzhiev ◽  
V. S. Gerdzhikov ◽  
M. I. Ivanov
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Hamiltonian Structures

An extension of the coupled derivative nonlinear Schrödinger hierarchy

2018 ◽  
Vol 32 (02) ◽  
pp. 1850016
Author(s):  
Siqi Xu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Compatibility Condition ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Nonlinear Schrödinger ◽  
Matrix Spectral Problem

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

TWO SUPER-INTEGRABLE SYSTEMS AND ASSOCIATED SUPER-HAMILTONIAN STRUCTURES

2009 ◽  
Vol 23 (27) ◽  
pp. 3253-3264 ◽  
Author(s):  
QIU-LAN ZHAO ◽  
XIN-YUE LI ◽  
BAI-YING HE
Keyword(s):  
Integrable Systems ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Lie Superalgebras ◽  
Nonlinear Evolution Equations ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Zero Curvature

The super extensions of g-cKdV and mKdV integrable systems are proposed. Two hierarchies of super-integrable nonlinear evolution equations are found. In addition, making use of the super-trace identity, we construct the super-Hamiltonian structures of zero-curvature equations associated with Lie superalgebras.

Conservation laws and Hamiltonian structures of the generalized sine-Gordon hierarchy

2014 ◽  
Vol 28 (32) ◽  
pp. 1450248
Author(s):  
Fang Li ◽  
Bo Xue ◽  
Yan Li
Keyword(s):  
Evolution Equations ◽  
Gordon Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Nonlinear Evolution Equations ◽  
Conserved Quantities ◽  
Trace Identity ◽  
Sine Gordon Equation ◽  
Hamiltonian Structures ◽  
Matrix Spectral Problem

By introducing a 2 × 2 matrix spectral problem, a new hierarchy of nonlinear evolution equations is proposed. A typical equation in this hierarchy is the generalization of sine-Gordon equation. With the aid of trace identity, the Hamiltonian structures of the hierarchy are constructed. In addition, the infinite sequence of conserved quantities of the generalized sine-Gordon equation are obtained.

SOME NEW INTEGRABLE NONLINEAR EVOLUTION EQUATIONS AND THEIR INFINITELY MANY CONSERVATION LAWS

2010 ◽  
Vol 24 (19) ◽  
pp. 2077-2090 ◽  
Author(s):  
XIANGUO GENG ◽  
BO XUE
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Nonlinear Evolution Equations ◽  
Conserved Quantities ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Matrix Spectral Problem

A hierarchy of new nonlinear evolution equations associated with a 3×3 matrix spectral problem with two potentials is derived and its Hamiltonian structures are established with the aid of trace identity. The negative flow of the hierarchy is then discussed. A reduction of this hierarchy and its Hamiltonian structures are constructed. An infinite sequence of conserved quantities of several new soliton equations is obtained.

An integrable extension of TD hierarchy and generalized bi-Hamiltonian structures

2015 ◽  
Vol 29 (21) ◽  
pp. 1550116 ◽  
Author(s):  
Huan Liu ◽  
Xianguo Geng
Keyword(s):  
Spectral Parameter ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Nonlinear Evolution Equations ◽  
Conserved Quantities ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Parameter Expansion ◽  
Spectral Problems

A hierarchy of new nonlinear evolution equations associated with [Formula: see text] matrix spectral problems are proposed, which is a naturally integrable extension of the TD hierarchy. It is shown that all nonlinear evolution equations in the hierarchy have generalized bi-Hamiltonian structures with the help of the trace identity. Moreover, the infinite sequence of conserved quantities of the first nontrivial equation in the hierarchy is constructed by means of spectral parameter expansion.

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