Difference Equations: The Toda Lattice

1981 ◽  
pp. 140-158
Author(s):  
Gert Eilenberger
Keyword(s):  
Difference Equations ◽  
Toda Lattice

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 Toda Lattice Solutions of Differential-Difference Equations for Dissipative Systems

1999 ◽  
Vol 68 (3) ◽  
pp. 791-796 ◽  
Author(s):  
Yuji Igarashi ◽  
Katsumi Itoh ◽  
Ken Nakanishi
Keyword(s):  
Difference Equations ◽  
Toda Lattice ◽  
Dissipative Systems

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 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.

Exp-Function Method for N-Soliton Solutions of Nonlinear Differential-Difference Equations

2010 ◽  
Vol 65 (11) ◽  
pp. 924-934 ◽  
Author(s):  
Sheng Zhang ◽  
Hong-Qing Zhang
Keyword(s):  
Symbolic Computation ◽  
Difference Equations ◽  
Function Method ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Mathematical Physics ◽  
Mathematical Tool

In this paper, the exp-function method is generalized to construct N-soliton solutions of nonlinear differential-difference equations. With the aid of symbolic computation, we choose the Toda lattice to illustrate the validity and advantages of the generalized work. As a result, 1-soliton, 2-soliton, and 3-soliton solutions are obtained, from which the uniform formula of N-soliton solutions is derived. It is shown that the exp-function method may provide us with a straightforward and effective mathematical tool for generating N-soliton solutions of nonlinear differential-difference equations in mathematical physics.

Change of the Time for the Toda Lattice

2001 ◽  
Vol 8 (Supplement) ◽  
pp. 278
Author(s):  
A V TSIGANOV
Keyword(s):  
Toda Lattice

On the solvability of the degenerate Noetherian difference-algebraic boundary value problem

2019 ◽  
Vol 33 ◽  
pp. 204-217
Author(s):  
Sergei Chuiko ◽  
Yaroslav Kalinichenko ◽  
Nikita Popov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Nonlinear Oscillations ◽  
Bounded Operator ◽  
Boundary Value ◽  
Homogeneous Part ◽  
Solvability Conditions ◽  
Engineering Control ◽  
Generalized Green Operator

The original conditions of solvability and the scheme of finding solutions of a linear Noetherian difference-algebraic boundary-value problem are proposed in the article, while the technique of pseudoinversion of matrices by Moore-Penrose is substantially used. The problem posed in the article continues to study the conditions for solvability of linear Noetherian boundary value problems given in the monographs of A.M. Samoilenko, A.V. Azbelev, V.P. Maximov, L.F. Rakhmatullina and A.A. Boichuk. The study of differential-algebraic boundary-value problems is closely related to the investigation of boundary-value problems for difference equations, initiated in the works of A.A. Markov, S.N. Bernstein, Y.S. Bezikovych, O.O. Gelfond, S.L. Sobolev, V.S. Ryabenkyi, V.B. Demidovych, A. Halanai, G.I. Marchuk, A.A. Samarskyi, Yu.A. Mytropolskyi, D.I. Martyniuk, G.M. Vainiko, A.M. Samoilenko and A.A. Boichuk. On the other hand, the study of boundary-value problems for difference equations is related to the study of differential-algebraic boundary-value problems initiated in the papers of K. Weierstrass, N.N. Lusin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, N.A. Perestiyk, V.P. Yakovets, A.A. Boichuk, A. Ilchmann and T. Reis. The study of differential-algebraic boundary value problems is also associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, control theory, motion stability theory. The general case of a linear bounded operator corresponding to the homogeneous part of a linear Noetherian difference-algebraic boundary value problem has no inverse is investigated. The generalized Green operator of a linear difference-algebraic boundary value problem is constructed in the article. The relevance of the study of solvability conditions, as well as finding solutions of linear Noetherian difference-algebraic boundary-value problems, is associated with the widespread use of difference-algebraic boundary-value problems obtained by linearizing nonlinear Noetherian boundary-value problems for systems of ordinary differential and difference equations. Solvability conditions are proposed, as well as the scheme of finding solutions of linear Noetherian difference-algebraic boundary value problems are illustrated in detail in the examples.

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