Advanced Mathematical Models & Applications

An initial boundary-value problem with zero boundary condition for an infinite system of nonlinear evolution equations is considered. First, the scattering problem for a second-order difference equation, which is related to the problem under consideration, was considered. We prove the existence and uniqueness of a rapidly decreasing solution and determine a class of initial data that guarantees the existence of a rapidly decreasing solution. Then, the scattering problem is considered for a second-order difference equation whose coefficients are solutions to a nonlinear equation. A law of variation of scattering data with time has been found. Using the variation of the scattering data and by the inverse spectral method, an algorithm for constructing the solution is obtained.