scholarly journals Discrete canonical system and non-Abelian Toda lattice: Bäcklund–Darboux transformation, Weyl functions, and explicit solutions

2007 ◽  
Vol 280 (5-6) ◽  
pp. 631-653 ◽  
Author(s):  
A. L. Sakhnovich
Keyword(s):  
Darboux Transformation ◽  
Toda Lattice ◽  
Canonical System ◽  
Explicit Solutions ◽  
Weyl Functions

A generalized integrable hierarchy related to the relativistic Toda lattice: Hamiltonian structure, Darboux transformation, soliton solution and conservation law

2021 ◽  
Author(s):  
Zhiguo Xu
Keyword(s):  
Darboux Transformation ◽  
Hamiltonian Structure ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Integrable Hierarchy ◽  
Integrable Lattice ◽  
Pass Through ◽  
Matrix Spectral Problem ◽  
Relativistic Toda Lattice ◽  
Toda Lattice Hierarchy

Starting from a more generalized discrete [Formula: see text] matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the [Formula: see text]-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete [Formula: see text]-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.

Darboux transformation and two-dimensional Toda lattice

1983 ◽  
Vol 23 (4) ◽  
pp. 2441-2446 ◽  
Author(s):  
V. B. Matveev ◽  
M. A. Salle
Keyword(s):  
Darboux Transformation ◽  
Toda Lattice ◽  
Two Dimensional

Explicit Solutions and Conservation Laws of a Coupled Burgers’ Equation

2017 ◽  
Vol 72 (9) ◽  
pp. 789-793
Author(s):  
Bo Xue ◽  
Fang Li ◽  
Yihao Li ◽  
Mingming Sun
Keyword(s):  
Conservation Laws ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Burgers Equation ◽  
Integrable Equation ◽  
Explicit Solutions ◽  
Spectral Problems ◽  
Coupled Burgers Equation

AbstractBased on the gauge transformation between the corresponding 3×3 matrix spectral problems, N-fold Darboux transformation for a coupled Burgers’ equation is constructed. Considering the N=1 case of the derived Darboux transformation, explicit solutions for the coupled Burgers’ equation are given and their figures are plotted. Moreover, conservation laws of this integrable equation are deduced.

scholarly journals A Relativistic Toda Lattice Hierarchy, Discrete Generalized (m,2N−m)-Fold Darboux Transformation and Diverse Exact Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2315
Author(s):  
Meng-Li Qin ◽  
Xiao-Yong Wen ◽  
Manwai Yuen
Keyword(s):  
Exact Solutions ◽  
Darboux Transformation ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Lattice System ◽  
Arbitrary Parameter ◽  
Rational Solutions ◽  
Limit States ◽  
Hybrid Solutions ◽  
Relativistic Toda Lattice

This paper investigates a relativistic Toda lattice system with an arbitrary parameter that is a very remarkable generalization of the usual Toda lattice system, which may describe the motions of particles in lattices. Firstly, we study some integrable properties for this system such as Hamiltonian structures, Liouville integrability and conservation laws. Secondly, we construct a discrete generalized (m,2N−m)-fold Darboux transformation based on its known Lax pair. Thirdly, we obtain some exact solutions including soliton, rational and semi-rational solutions with arbitrary controllable parameters and hybrid solutions by using the resulting Darboux transformation. Finally, in order to understand the properties of such solutions, we investigate the limit states of the diverse exact solutions by using graphic and asymptotic analysis. In particular, we discuss the asymptotic states of rational solutions and exponential-and-rational hybrid solutions graphically for the first time, which might be useful for understanding the motions of particles in lattices. Numerical simulations are used to discuss the dynamics of some soliton solutions. The results and properties provided in this paper may enrich the understanding of nonlinear lattice dynamics.

Sign in / Sign up

Export Citation Format

Share Document