New Exact Solutions of a Relativistic Toda Lattice System

2012 ◽  
Vol 29 (9) ◽  
pp. 094101 ◽  
Author(s):  
M. T. Darvishi ◽  
F. Khani
Keyword(s):  
Exact Solutions ◽  
Toda Lattice ◽  
Lattice System ◽  
New Exact Solutions ◽  
Relativistic Toda Lattice

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.

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.

APPROXIMATE SOLUTIONS OF NONLINEAR DIFFERENTIAL DIFFERENCE EQUATIONS

2009 ◽  
Vol 06 (04) ◽  
pp. 569-583 ◽  
Author(s):  
M. A. ABDOU
Keyword(s):  
Exact Solutions ◽  
Difference Equations ◽  
Homotopy Perturbation Method ◽  
Toda Lattice ◽  
Approximate Solutions ◽  
Mathematical Physics ◽  
Nonlinear Difference Equations ◽  
Science And Engineering ◽  
Homotopy Perturbation ◽  
Relativistic Toda Lattice

An extend of He's homotopy perturbation method (HPM) is used for finding a new approximate and exact solutions of nonlinear difference differential equations arising in mathematical physics. To illustrate the effectiveness and the advantage of the proposed method, two models of nonlinear difference equations of special interest in physics are chosen, namely, Ablowitz–Ladik lattice equations and Relativistic Toda lattice difference equations. Comparisons are made between the results of the proposed method and exact solutions. The results show that the HPM is a attracted method in solving the differential difference equations (DDEs). The proposed method will become a much more interesting method for solving nonlinear DDEs in science and engineering.

scholarly journals Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Khaled A. Gepreel ◽  
Taher A. Nofal ◽  
Fawziah M. Alotaibi
Keyword(s):  
Solitary Wave ◽  
Exact Solutions ◽  
Difference Equations ◽  
Toda Lattice ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Discrete Nonlinear Schrödinger Equation ◽  
Nonlinear Schrödinger ◽  
Saturable Nonlinearity ◽  
Relativistic Toda Lattice

We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

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