Exact solitons in the nonlocal Gordon equation

TL;DR

This paper derives exact monotonic solitons for a nonlocal Gordon equation with a specific kernel and introduces an inverse method to identify nonlinearities that admit such solutions, also analyzing related equations for kink solutions.

Contribution

It presents a novel inverse method to find nonlinearities allowing exact solitons in the nonlocal Gordon equation and studies related equations for kink solutions.

Findings

Exact monotonic solitons found for the nonlocal Gordon equation.

An inverse method characterizing nonlinearities with soliton solutions.

Existence of a 4π-kink and nonexistence of 2π-kinks in related equations.

Abstract

We find exact monotonic solitons in the nonlocal Gordon equation u_{tt}=J*u-u-f(u), in the case J(x)=1/2 e^{-|x|}. To this end we come up with an inverse method, which gives a representation of the set of nonlinearities admitting such solutions. We also study u''''+{\l}u''-sin u=0, which arises from the above when we write it in traveling wave coordinates and pass to a certain limit. For this equation we find an exact 4\pi-kink and show the nonexistence of 2\pi-kinks, using the analytic continuation method of Amick and McLeod.