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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.

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