Exact solitons in the nonlocal Gordon equation

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.