Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity
We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. 
As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.