DARBOUX TRANSFORMATION OF THE MODIFIED TODA LATTICE EQUATION

2006 ◽  
Vol 20 (11) ◽  
pp. 641-648 ◽  
Author(s):  
XI-XIANG XU ◽  
HONG-XIANG YANG ◽  
YE-PENG SUN
Keyword(s):  
Spectral Problem ◽  
Soliton Solution ◽  
Darboux Transformation ◽  
Toda Lattice ◽  
Lattice Equation ◽  
Matrix Spectral Problem ◽  
Toda Lattice Equation

A modified Toda lattice equation associated with a properly discrete matrix spectral problem is introduced. Darboux transformation for the resulting lattice equation is constructed. As an application, the soliton solution for the Toda lattice equation is explicitly given.

A GENERALIZATION OF TODA LATTICES AND THEIR BI-HAMILTONIAN STRUCTURES

2012 ◽  
Vol 26 (13) ◽  
pp. 1250078 ◽  
Author(s):  
XIANGUO GENG ◽  
FANG LI ◽  
BO XUE
Keyword(s):  
Spectral Problem ◽  
Fourth Order ◽  
Difference Equations ◽  
Toda Lattice ◽  
Trace Identity ◽  
Lattice Equation ◽  
Hamiltonian Structures ◽  
Typical Member ◽  
Toda Lattice Equation ◽  
Toda Lattices

A hierarchy of new nonlinear differential-difference equations associated with fourth-order discrete spectral problem is proposed, in which a typical member is a generalization of the Toda lattice equation. The bi-Hamiltonian structures for this hierarchy are obtained with the help of trace identity.

scholarly journals Tropical limit of matrix solitons and entwining Yang–Baxter maps

2020 ◽  
Vol 110 (11) ◽  
pp. 3015-3051
Author(s):  
Aristophanes Dimakis ◽  
Folkert Müller-Hoissen
Keyword(s):  
Soliton Solution ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Baxter Equation ◽  
Kp Equation ◽  
Lattice Equation ◽  
Soliton Interactions ◽  
The Matrix ◽  
Tropical Limit ◽  
Toda Lattice Equation

Abstract We consider a matrix refactorization problem, i.e., a “Lax representation,” for the Yang–Baxter map that originated as the map of polarizations from the “pure” 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang–Baxter equation, but satisfies a mixed version of the Yang–Baxter equation together with the Yang–Baxter map. Such maps have been called “entwining Yang–Baxter maps” in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is not in general a Yang–Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang–Baxter equation holds, by exploring the pure 3-soliton solution in the “tropical limit,” where the 3-soliton interaction decomposes into 2-soliton interactions. Here, this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang–Baxter maps as in the KP case, indicating a kind of universality.

scholarly journals Explicit Solutions and Conservation Laws for a New Integrable Lattice Hierarchy

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Qianqian Yang ◽  
Qiulan Zhao ◽  
Xinyue Li
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Spectral Problem ◽  
Darboux Transformation ◽  
Hamiltonian Structure ◽  
Explicit Solutions ◽  
Liouville Integrability ◽  
Lattice Equation ◽  
Integrable Lattice ◽  
Matrix Spectral Problem

An integrable lattice hierarchy is derived on the basis of a new matrix spectral problem. Then, some properties of this hierarchy are shown, such as the Liouville integrability, the bi-Hamiltonian structure, and infinitely many conservation laws. After that, the Darboux transformation of the first integrable lattice equation in this hierarchy is constructed. Eventually, the explicitly exact solutions of the integrable lattice equation are investigated via graphs.

A KIND OF EXPLICIT QUASI-PERIODIC SOLUTION AND ITS LIMIT FOR THE TODA LATTICE EQUATION

2008 ◽  
Vol 22 (08) ◽  
pp. 547-553 ◽  
Author(s):  
Y. C. HON ◽  
ENGUI FAN ◽  
ZHENYUN QIN
Keyword(s):  
Periodic Solution ◽  
Asymptotic Property ◽  
Soliton Solution ◽  
Theta Function ◽  
Toda Lattice ◽  
Hirota Bilinear Method ◽  
Lattice Equation ◽  
Bilinear Method ◽  
Toda Lattice Equation ◽  
Hirota Bilinear

Based on the Riemann theta function, the Hirota bilinear method is extended to directly construct a kind of explicit quasi-periodic solution of the Toda lattice equation. The asymptotic property of the quasi-periodic solution is analyzed in detail. It is of interest to see that the well-known soliton solution can be obtained as a limit of the quasi-periodic solution.

N-SOLITON SOLUTIONS AND CONSERVATION LAWS OF THE MODIFIED TODA LATTICE EQUATION

2012 ◽  
Vol 26 (06) ◽  
pp. 1150032 ◽  
Author(s):  
XIAO-YONG WEN
Keyword(s):  
Conservation Laws ◽  
Darboux Transformation ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Lax Representation ◽  
Lattice Equation ◽  
Interaction Behavior ◽  
Toda Lattice Equation

The modified Toda lattice equation is investigated via Darboux transformation (DT) technique, the N-fold DT is constructed based on its Lax representation. The N-soliton solutions are also derived via the resulting N-fold DT. Soliton structures and interaction behavior of those solutions are shown graphically. Finally, the infinitely many conservation laws for that system are given.

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