Exact Solutions of One Nonlinear Countable-Dimensional System of Integro-Differential Equations

Журнал Средневолжского математического общества. 2023. Vol. 25. No. 1. P. 542–533.

A. E. Rassadin

In the present paper, a nonlinear countable-dimensional system of integrodifferential equations is investigated, whose vector of unknowns is a countable set of functions of two variables. These variables are interpreted as spatial coordinate and time. The nonlinearity of this system is constructed from two simultaneous convolutions: first convolution is in the sense of functional analysis and the second one is in the sense of linear space of double-sided sequences. The initial condition for this system is a doublesided sequence of functions of one variable defined on the entire real axis. The system itself can be written as a single abstract equation in the linear space of double-sided sequences. As the system may be resolved with respect to the time derivative, it may be presented as a dynamical system. The solution of this abstract equation can be interpreted as an approximation of the solution of a nonlinear integro-differential equation, whose unknown function depends not only on time, but also on two spatial variables. General representation for exact solution of system under study is obtained in the paper. Also two kinds of particular examples of exact solutions are presented. The first demonstrates oscillatory spatio-temporal behavior, and the second one shows monotone in time behavior. In the paper typical graphs of the first components of these solutions are plotted. Moreover, it is demonstrated that using some procedure one can generate countable set of new exact system’s solutions from previously found solutions. From radio engineering point of view this procedure just coincides with procedure of upsampling in digital signal processing.

Keywords: производящая функция, преобразование Фурье, Laurent series, Cauchy problem, Fourier transform, modified Bessel functions, модифицированные функции Бесселя, generating function, Bessel functions of the first kind, задача Коши, ряд Лорана, функции Бесселя первого рода