This article is cited in 4 papers

Partial Differential Equations

Existence of bounded soliton solutions in the problem of longitudinal oscillations of an elastic infinite rod in a field with a nonlinear potential of general form

A. L. Beklaryan^a, L. A. Beklaryan<sup>b

a</sup> Central Economics and Mathematics Institute, Russian Academy of Sciences, 117418, Moscow, Russia
^b National Research University Higher School of Economics, 101000, Moscow, Russia

Abstract: The existence of a family of bounded soliton solutions for a finite-difference analogue of the wave equation with a general nonlinear potential is proved. The proof is based on a formalism establishing a one-to-one correspondence between soliton solutions of an infinite-dimensional dynamical system and solutions of a family of functional differential equations of the pointwise type. A key point in the proof of the existence of bounded soliton solutions is a theorem on the existence and uniqueness of soliton solutions in the case of a quasilinear potential. Another important circumstance for the considered class of systems of equations is that they have a number of symmetries due to the low dimension (one-dimensionality) of the space at each lattice point.

Key words: wave equation, soliton solutions, nonlinear potential.

UDC: 517.9

Received: 24.12.2021
Revised: 15.01.2022
Accepted: 15.01.2022

DOI: 10.31857/S0044466922060035

 English version: Computational Mathematics and Mathematical Physics, 2022, 62:6, 904–919

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