On the existence of soliton solutions for systems with a polynomial potential and their numerical realization

Abstract

The problem of existence of soliton solutions (solutions of the traveling wave type) for the Korteweg-de Vries equation with a polynomial potential is considered on the basis of the approach within which the presence of a one-to-one correspondence of such solutions with solutions of the induced functional differential equation of pointwise type is demonstrated. On this path, conditions for the existence and uniqueness of solutions of the traveling wave type, with the growth restrictions both in time and in space, arise. It is very important that the conditions for the existence of a traveling wave solution are formed in terms of the right-hand side of the equation and the characteristics of the traveling wave, without using either the linearization and spectral properties of the corresponding equation in variations. Conditions for the existence of periodic soliton solutions are considered separately, and the possibility of transition from systems with a quasilinear potential to systems with a polynomial potential with conservation of corresponding existence theorems is demonstrated. Numerical implementation of such solutions is given.