scholarly journals Exact Solutions and Numerical Simulation of the Discrete Sawada–Kotera Equation

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 131
Author(s):  
Aleksandr Zemlyanukhin ◽  
Andrey Bochkarev
Keyword(s):  
Numerical Simulation ◽  
Difference Equation ◽  
Phase Velocity ◽  
Jacobi Elliptic Function ◽  
Initial Condition ◽  
Series Method ◽  
Localized Solutions ◽  
Exponential Generating Function ◽  
Differential Difference Equation ◽  
Negative Phase Velocity

We investigated an integrable five-point differential-difference equation called the discrete Sawada–Kotera equation. On the basis of the geometric series method, a new exact soliton-like solution of the equation is obtained that propagates with positive or negative phase velocity. In terms of the Jacobi elliptic function, a class of new exact periodic solutions is constructed, in particular stationary ones. Using an exponential generating function for Catalan numbers, Cauchy’s problem with the initial condition in the form of a step is solved. As a result of numerical simulation, the elasticity of the interaction of exact localized solutions is established.

On dichotomic maps for a class of differential-difference equations

1991 ◽  
Vol 117 (3-4) ◽  
pp. 317-328 ◽  
Author(s):  
L. A. V. Carvalho ◽  
K. L. Cooke
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Difference Equations ◽  
Positive Definite ◽  
Initial Condition ◽  
Null Solution ◽  
Differential Difference Equation

SynopsisStability and asymptotic stability of the null solution of the differential-difference equation (E)x′(t) = f(x(t), x(t − r)), f: RNxRN → RN, f(0, 0) = 0, are studied by means of an extension of the Liapunov–Razumikhin method. Let V: RN → R be a differentiate map, let C = C(+ −r, 0=, RN), and let x(t, ψ) denote the solution of (E) with initial condition ψ in C at t = 0. For t ≧ 0 let xt(ψ) be defined by xt,(ψ)(θ) = x(t + θ, ψ), −r ≦θ ≦0. Let V′ (ψ) be the variation of V along the solution x(t, ψ). We say that V is dichotomic with respect to (E) if there exist T ≧0 and Ω, a neighbourhood of the origin in C, such that if ψ is in the closure of the set where V′ (xT(ψ)) >; 0, then V(x(T, ψ)) ≦ V(x(s, ψ)) for some s, −r ≦ sT. It is proved that if V is positive definite, continuously differentiable, and dichotomic, then the null solution of (E) is stable. A concept of strict dichotomic map is introduced and used to prove asymptotic stability. A number of examples are given to illustrate the applications of the method.

Sign in / Sign up

Export Citation Format

Share Document