Comment to: “Two hierarchies of lattice soliton equations associated with a new discrete eigenvalue problem and Darboux transformation” [Phys. Lett. A 338 (2005) 117]

2006 ◽  
Vol 350 (5-6) ◽  
pp. 419-420 ◽  
Author(s):  
A.K. Svinin
Keyword(s):  
Eigenvalue Problem ◽  
Darboux Transformation ◽  
Discrete Eigenvalue ◽  
Soliton Equations

A Hierarchy of Lattice Soliton Equations Associated with a New Discrete Eigenvalue Problem and Darboux Transformations

2015 ◽  
Vol 16 (7-8) ◽  
pp. 301-306 ◽  
Author(s):  
Ning Zhang ◽  
Tiecheng Xia
Keyword(s):  
Eigenvalue Problem ◽  
Exact Solutions ◽  
Spectral Parameter ◽  
Darboux Transformation ◽  
Lattice Models ◽  
Darboux Transformations ◽  
Discrete Eigenvalue ◽  
Negative Power ◽  
Soliton Equations ◽  
Gauge Transformations

AbstractBy considering a new discrete isospectral eigenvalue problem, a hierarchy of integrable positive and negative lattice models is derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And the equation in the resulting hierarchy is integrable in Liouville sense. Further, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.

Eigen Solutions of Grouped Turbo Blades Solved by the Generalized Differential Quadrature Method

2001 ◽  
Author(s):  
J. H. Kuang ◽  
M. H. Hsu
Keyword(s):  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Eigenvalue Problems ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Discrete Eigenvalue ◽  
Generalized Differential Quadrature Method ◽  
Generalized Differential ◽  
Eigen Solutions

The eigenvalue problems of grouped turbo blades were numerically formulated by using the generalized differential quadrature method (GDQM). Different boundary approaches accompanying the GDQM to transform the partial differential equations of grouped turbo blades into a discrete eigenvalue problem are discussed. Effects of the number of sample points and the different boundary approaches on the accuracy of the calculated natural frequencies are also studied. Numerical results demonstrated the validity and the efficiency of the GDQM in treating this type of problem.

Eigensolutions of Grouped Turbo Blades Solved by the Generalized Differential Quadrature Method

2002 ◽  
Vol 124 (4) ◽  
pp. 1011-1017 ◽  
Author(s):  
J. H. Kuang ◽  
M. H. Hsu
Keyword(s):  
Partial Differential Equations ◽  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Eigenvalue Problems ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Discrete Eigenvalue ◽  
Generalized Differential Quadrature Method ◽  
Generalized Differential

The eigenvalue problems of grouped turbo blades were numerically formulated by using the generalized differential quadrature method (GDQM). Different boundary approaches accompanying the GDQM to transform the partial differential equations of grouped turbo blades into a discrete eigenvalue problem are discussed. Effects of the number of sample points and the different boundary approaches on the accuracy of the calculated natural frequencies are also studied. Numerical results demonstrated the validity and the efficiency of the GDQM in treating this type of problem.

Darboux Transformation for Coupled Non-Linear Schrödinger Equation and Its Breather Solutions

2017 ◽  
Vol 72 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Lili Feng ◽  
Fajun Yu ◽  
Li Li
Keyword(s):  
Schrödinger Equation ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Schrodinger Equation ◽  
Soliton Solutions ◽  
Soliton Equations ◽  
Spectral Problems ◽  
Non Linear ◽  
And Mathematics

AbstractStarting from a 3×3 spectral problem, a Darboux transformation (DT) method for coupled Schrödinger (CNLS) equation is constructed, which is more complex than 2×2 spectral problems. A scheme of soliton solutions of an integrable CNLS system is realised by using DT. Then, we obtain the breather solutions for the integrable CNLS system. The method is also appropriate for more non-linear soliton equations in physics and mathematics.

THE GENERALIZED MULTI-COMPONENT AKNS HIERARCHY AND N-FOLD DARBOUX TRANSFORMATION

2006 ◽  
Vol 20 (25) ◽  
pp. 1575-1589 ◽  
Author(s):  
HONG-XIANG YANG ◽  
DAO-LIN WANG ◽  
CHANG-SHENG LI
Keyword(s):  
Spectral Problem ◽  
Darboux Transformation ◽  
Soliton Solutions ◽  
Soliton Equation ◽  
Lax Pairs ◽  
Akns Hierarchy ◽  
Soliton Equations ◽  
Gauge Transformations ◽  
Set Up ◽  
Liouville Integrable

Starting from a 3×3 spectral problem, by using the Tu scheme, a hierarchy of generalized multi-component AKNS soliton equations are derived. It is shown that each equation in the resulting hierarchy is Liouville integrable. With the help of gauge transformations of the Lax pairs, an N-fold Darboux transformation (DT) with multi-parameters for the spectral problem is set up. For application, the soliton solutions of the first nonlinear soliton equation are explicitly given.

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