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

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.