A HIERARCHY OF HAMILTONIAN EQUATIONS ASSOCIATED WITH THE MODIFIED JAULENT–MIODEK HIERARCHY

By considering a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is presented.

Download Full-text

A FOUR-BY-FOUR MATRIX EIGENVALUE PROBLEM, RELATED HIERARCHY OF LAX INTEGRABLE EQUATIONS AND THEIR HAMILTONIAN STRUCTURES

International Journal of Modern Physics B ◽

10.1142/s0217979208048735 ◽

2008 ◽

Vol 22 (23) ◽

pp. 4027-4040 ◽

Cited By ~ 3

Author(s):

XI-XIANG XU ◽

HONG-XIANG YANG ◽

WEI-LI CAO

Keyword(s):

Eigenvalue Problem ◽

Evolution Equations ◽

Trace Identity ◽

Liouville Integrability ◽

Matrix Eigenvalue Problem ◽

Hamiltonian Structures ◽

Hamiltonian Equations ◽

Integrable Equations ◽

Matrix Eigenvalue ◽

Integrable Evolution

Starting from a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is proved.

Download Full-text

Three-Component Generalisation of Burgers Equation and Its Bi-Hamiltonian Structures

Zeitschrift für Naturforschung A ◽

10.1515/zna-2016-0493 ◽

2017 ◽

Vol 72 (5) ◽

pp. 469-475

Author(s):

Wei Liu ◽

Xianguo Geng ◽

Bo Xue

Keyword(s):

Eigenvalue Problem ◽

Conservation Laws ◽

Spectral Parameter ◽

Burgers Equation ◽

Trace Identity ◽

Matrix Eigenvalue Problem ◽

Hamiltonian Structures ◽

Curvature Equation ◽

Matrix Eigenvalue ◽

Zero Curvature

AbstractA hierarchy of three-component generalisation of Burgers equation, which is associated with a 3×3 matrix eigenvalue problem, is generated by using the zero-curvature equation. By means of the trace identity, the bi-Hamiltonian structures of this hierarchy are constructed. Moreover, the infinite conservation laws for the hierarchy are obtained with the aid of spectral parameter expansion.

Download Full-text

A New Hierarchy of Evolution Equations and its Integrable System

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.143-144.1200 ◽

2010 ◽

Vol 143-144 ◽

pp. 1200-1203

Author(s):

Shu Juan Yuan ◽

Mei Xia Chen

Keyword(s):

Eigenvalue Problem ◽

Integrable System ◽

Evolution Equations ◽

Second Order ◽

Matrix Eigenvalue Problem ◽

Lax Pairs ◽

Matrix Eigenvalue ◽

Hamilton System ◽

Finite Dimensional ◽

Order Matrix

The second-order matrix eigenvalue problem is discussed by means of the nonlinearization of the Lax pairs,then based on the Bargmann constraint between the potential and the eigenfunctions,a new finite-dimensional Hamilton system is abtained by nonlinearization of the eigenvalue problem and the involutive solutions of the evolution equations are abtained

Download Full-text

A note on a perturbation theorem for the matrix eigenvalue problem

Numerische Mathematik ◽

10.1007/bf02163233 ◽

1969 ◽

Vol 13 (2) ◽

pp. 152-153 ◽

Cited By ~ 1

Author(s):

John E. Osborn

Keyword(s):

Eigenvalue Problem ◽

Matrix Eigenvalue Problem ◽

Perturbation Theorem ◽

Matrix Eigenvalue ◽

The Matrix

Download Full-text

A Matrix Eigenvalue Problem (G. Efroymson, A. Steger and S. Steinberg)

SIAM Review ◽

10.1137/1022012 ◽

1980 ◽

Vol 22 (1) ◽

pp. 99-100

Author(s):

T. Sekiguchi ◽

N. Kimura

Keyword(s):

Eigenvalue Problem ◽

Matrix Eigenvalue Problem ◽

Matrix Eigenvalue

Download Full-text

On the Spectrum of a Nonlinear Two Parameter Matrix Eigenvalue Problem

Applications of Nonlinear Analysis - Springer Optimization and Its Applications ◽

10.1007/978-3-319-89815-5_13 ◽

2018 ◽

pp. 387-402 ◽

Cited By ~ 1

Author(s):

Michael Gil’

Keyword(s):

Eigenvalue Problem ◽

Matrix Eigenvalue Problem ◽

Matrix Eigenvalue ◽

Parameter Matrix ◽

Two Parameter

Download Full-text

An extension of the coupled derivative nonlinear Schrödinger hierarchy

Modern Physics Letters B ◽

10.1142/s0217984918500161 ◽

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.

Download Full-text

The study of numerical analysis of matrix eigenvalue problem in China

Contemporary Mathematics - Computational Mathematics in China ◽

10.1090/conm/163/01550 ◽

1994 ◽

pp. 71-76

Author(s):

Er Xiong Jiang

Keyword(s):

Numerical Analysis ◽

Eigenvalue Problem ◽

Matrix Eigenvalue Problem ◽

Matrix Eigenvalue

Download Full-text

Infinite Matrices, Ritz Theorem and Bound State Problems

Zeitschrift für Naturforschung A ◽

10.1515/zna-1975-0221 ◽

1975 ◽

Vol 30 (2) ◽

pp. 256-261 ◽

Cited By ~ 2

Author(s):

A. K. Mitra

Keyword(s):

Eigenvalue Problem ◽

Convenient Method ◽

Bound State ◽

Wave Functions ◽

Infinite Matrix ◽

Variational Technique ◽

Matrix Eigenvalue Problem ◽

Schroedinger Operator ◽

Matrix Eigenvalue ◽

Finite Matrix

Abstract The straight forward application of the Ritz variational technique has been shown to be a very convenient method for obtaining numerically the first few discrete eigenvalues of the Schroedinger operator with certain special types of potentials. This method solves essentially the (finite) matrix eigenvalue problem obtained by truncating the infinite matrix representing the Schroedinger operator with respect to the Coulomb wave functions. The Ritz theorem justifies the validity of this truncation procedure.

Download Full-text