Scalar Lax-pairs and soliton solutions of the generalized periodic Toda lattice

1992 ◽  
pp. 205-240
Author(s):  
A. N. Leznov ◽  
M. V. Saveliev
Keyword(s):  
Toda Lattice ◽  
Soliton Solutions ◽  
Lax Pairs

On the Full-Discrete Extended Generalised q-Difference Toda System

2017 ◽  
Vol 72 (8) ◽  
pp. 703-709
Author(s):  
Chuanzhong Li ◽  
Anni Meng
Keyword(s):  
Difference Equation ◽  
Hamiltonian Structure ◽  
Soliton Solutions ◽  
Toda Equation ◽  
Lax Pairs ◽  
Toda System ◽  
Toda Hierarchy

AbstractIn this paper, we construct a full-discrete integrable difference equation which is a full-discretisation of the generalised q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of an extended generalised full-discrete q-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the extended full-discrete generalised q-Toda hierarchy are given.

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.

Exact solutions and aysmptotic state analysis of a modified Toda lattice system with a perturbation parameter

2021 ◽  
Author(s):  
Meng-Li Qin ◽  
Xiao-Yong Wen ◽  
Cui-Lian Yuan
Keyword(s):  
Asymptotic Analysis ◽  
Exact Solutions ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Perturbation Parameter ◽  
Lattice System ◽  
Rational Solutions ◽  
Limit States ◽  
Mixed Solutions ◽  
Analysis Technique

Under consideration is a modified Toda lattice system with a perturbation parameter, which may describe the particle motion in a lattice. With the aid of symbolic computation Maple, the discrete generalized [Formula: see text]-fold Darboux transformation (DT) of this system is constructed for the first time. Different types of exact solutions are derived by applying the resulting DT through choosing different [Formula: see text]. Specifically, standard soliton solutions, rational solutions and their mixed solutions are given via the [Formula: see text]-fold DT, [Formula: see text]-fold DT and [Formula: see text]-fold DT, respectively. Limit states of various exact solutions are analyzed via the asymptotic analysis technique. Compared with the known results, we find that the asymptotic states of mixed solutions of standard soliton and rational solutions are consistent with the asymptotic analysis results of solitons and rational solutions alone. Soliton interaction and propagation phenomena are shown graphically. Numerical simulations are used to explore relevant soliton dynamical behaviors. These results and properties might be helpful for understanding lattice dynamics.

CIRCUIT IMPLEMENTATIONS OF SOLITON SYSTEMS

1999 ◽  
Vol 09 (04) ◽  
pp. 571-590 ◽  
Author(s):  
ANDREW C. SINGER ◽  
ALAN V. OPPENHEIM
Keyword(s):  
Analog Circuits ◽  
Toda Lattice ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Analytic Solutions ◽  
Complex Signal ◽  
Kdv Equations ◽  
Exact Analytic Solutions ◽  
Signal Processors ◽  
Soliton Collisions

Recently, a large class of nonlinear systems which possess soliton solutions has been discovered for which exact analytic solutions can be found. Solitons are eigenfunctions of these systems which satisfy a form of superposition and display rich signal dynamics as they interact. In this paper, we view solitons as signals and consider exploiting these systems as specialized signal processors which are naturally suited to a number of complex signal processing tasks. New circuit models are presented for two soliton systems, the Toda lattice and the discrete-KdV equations. These analog circuits can generate and process soliton signals and can be used as multiplexers and demultiplexers in a number of potential soliton-based wireless communication applications discussed in [Singer et al.]. A hardware implementation of the Toda lattice circuit is presented, along with a detailed analysis of the dynamics of the system in the presence of additive Gaussian noise. This circuit model appears to be the first such circuit sufficiently accurate to demonstrate true overtaking soliton collisions with a small number of nodes. The discrete-KdV equation, which was largely ignored for having no prior electrical or mechanical analog, provides a convenient means for processing discrete-time soliton signals.

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.

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.

Multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation via Hirota’s bilinear method

2014 ◽  
Vol 92 (3) ◽  
pp. 184-190 ◽  
Author(s):  
Sheng Zhang ◽  
Dong Liu
Keyword(s):  
Variable Coefficient ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Hirota’S Bilinear Method ◽  
Lattice Equation ◽  
Bilinear Method ◽  
Dimensional Variable ◽  
Toda Lattice Equation ◽  
Multisoliton Solutions

In this paper, Hirota’s bilinear method is extended to construct multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation. As a result, new and more general one-soliton, two-soliton, and three-soliton solutions are obtained, from which the uniform formula of the N-soliton solution is derived. It is shown that Hirota’s bilinear method can be used for constructing multisoliton solutions of some other nonlinear differential-difference equations with variable coefficients.

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