Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

2008 ◽  
Vol 49 (6) ◽  
pp. 1399-1402 ◽  
Author(s):  
Luo Lin ◽  
Fan En-Gui
Keyword(s):  
Hamiltonian System ◽  
Conservation Laws ◽  
Spectral Problem

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.

scholarly journals BINARY NONLINEARIZATION OF THE SUPER AKNS SYSTEM

2008 ◽  
Vol 22 (04) ◽  
pp. 275-288 ◽  
Author(s):  
JINGSONG HE ◽  
JING YU ◽  
YI CHENG ◽  
RUGUANG ZHOU
Keyword(s):  
Hamiltonian System ◽  
Spectral Problem ◽  
Integrable Hamiltonian System ◽  
Integrals Of Motion ◽  
Finite Dimensional ◽  
Akns System ◽  
Hamiltonian Forms

We establish the binary nonlinearization approach of the spectral problem of the super AKNS system, and then use it to obtain the super finite-dimensional integrable Hamiltonian system in the supersymmetry manifold ℝ4N|2N. The super Hamiltonian forms and integrals of motion are given explicitly.

scholarly journals Conservation Laws and Self-Consistent Sources for an Integrable Lattice Hierarchy Associated with a Three-by-Three Discrete Matrix Spectral Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yu-Qing Li ◽  
Bao-Shu Yin
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Liouville Integrability ◽  
Hamiltonian Structures ◽  
Integrable Lattice ◽  
Self Consistent ◽  
Matrix Spectral Problem

A lattice hierarchy with self-consistent sources is deduced starting from a three-by-three discrete matrix spectral problem. The Hamiltonian structures are constructed for the resulting hierarchy. Liouville integrability of the resulting equations is demonstrated. Moreover, infinitely many conservation laws of the resulting hierarchy are obtained.

A HIERARCHY OF LIOUVILLE INTEGRABLE LATTICE EQUATION ASSOCIATED WITH A THREE-BY-THREE DISCRETE SPECTRAL PROBLEM AND ITS INFINITELY MANY CONSERVATION LAWS

2011 ◽  
Vol 25 (18) ◽  
pp. 2481-2492
Author(s):  
YU-QING LI ◽  
XI-XIANG XU
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Trace Identity ◽  
Lattice Equation ◽  
Hamiltonian Structures ◽  
Discrete Soliton ◽  
Soliton Equations ◽  
Integrable Lattice ◽  
Matrix Spectral Problem ◽  
Liouville Integrable

A discrete three-by-three matrix spectral problem is put forward and the corresponding discrete soliton equations are deduced. By means of the trace identity the Hamiltonian structures of the resulting equations are constructed, and furthermore, infinitely many conservation laws of the corresponding lattice system are obtained by a direct way.

scholarly journals A Soliton Hierarchy Associated with a Spectral Problem of 2nd Degree in a Spectral Parameter and Its Bi-Hamiltonian Structure

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Yuqin Yao ◽  
Shoufeng Shen ◽  
Wen-Xiu Ma
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Liouville Integrability ◽  
Hamiltonian Structures ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Matrix Spectral Problem

Associated withso~(3,R), a new matrix spectral problem of 2nd degree in a spectral parameter is proposed and its corresponding soliton hierarchy is generated within the zero curvature formulation. Bi-Hamiltonian structures of the presented soliton hierarchy are furnished by using the trace identity, and thus, all presented equations possess infinitely commuting many symmetries and conservation laws, which implies their Liouville integrability.

THE NEGATIVE AND POSITIVE INTEGRABLE COUPLING HIERARCHIES DERIVED FROM A FOUR BY FOUR MATRIX SPECTRAL PROBLEM

2010 ◽  
Vol 24 (30) ◽  
pp. 2955-2970
Author(s):  
XI-XIANG XU
Keyword(s):  
Hamiltonian System ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Hamiltonian Form ◽  
Liouville Integrability ◽  
Positive Power ◽  
Integrable Lattice ◽  
Lax Operator ◽  
Matrix Spectral Problem ◽  
Integrable Coupling

Discrete integrable coupling hierarchies of two existing integrable lattice families are derived from a four by four discrete matrix spectral problem. It is shown that the obtained integrable coupling hierarchies respectively corresponds to negative and positive power expansions of the Lax operator with respect to the spectral parameter. Then, the Hamiltonian form of the negative integrable coupling hierarchy is constructed by using the discrete variational identity. Finally, Liouville integrability of each obtained discrete Hamiltonian system is demonstrated.

Sign in / Sign up

Export Citation Format

Share Document