Abstract
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.
Highlights
The theory of integrable differential-difference equations (DDE), based on the study of symmetries, conservation laws and recursion operators, is actively developing at present
The geometric series method is designed to search for exact DDE solutions, and the solutions found are expressed in terms of the ratio of two polynomials in powers of the exponential function
A number of problems are solved for an integrable discrete Sawada–Kotera equation
Summary
The theory of integrable differential-difference equations (DDE), based on the study of symmetries, conservation laws and recursion operators, is actively developing at present. Nonlinear models of physical processes allow analytical research only on the basis of the continuum limit of the initial discrete equation [4]. This is due to the fact that the construction of exact solutions of nonlinear DDEs other than the Toda, Volterra and Ablowitz–Ladik lattices is difficult [5]. Most integrable lattices have the Korteweg–de Vries and nonlinear Schrodinger equations as a continuum limit [6,7]. As a result of the classification of five-point DDEs [8], integrable discretizations of the Sawada–Kotera and Kaup–Kupershmidt equations have been identified
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