Existence of bounded soliton solutions in the problem of longitudinal vibrations of an infinite elastic rod in a field with a strongly nonlinear potential

The existence of a family of bounded soliton solutions for a finite-difference wave equation with a quadratic potential is established. The proof is based on a formalism establishing a one-to-one correspondence between the soliton solutions of an infinite-dimensional dynamical system and the solutions of a family of functional differential equations of the pointwise type. A key point for the considered class of equations is also the existence of a number of symmetries.