Traveling waves for the nonlocal Gross-Pitaevskii equation: a min-max approach

We deal with the existence of finite-energy traveling wave solutions for the Gross-Pitaevskii equation in 1D with a general nonlocal interaction between particles and with nontrivial conditions at infinity. These kinds of solution correspond to dark solitons in the physical literature. In a previous work, A. de Laire and P. Mennuni proved the existence of a branch of finite-energy and orbitally stable dark solitons parametrized in terms of the momentum. In this way, the wave speed appears as a Lagrange multiplier, but there is no enough control to ensure that all subsonic speeds are reached. In a joint work with A. de Laire, we show that there exists a finite-energy traveling wave for almost (in the sense of the Lebesgue measure) all subsonic speed. Our approach, inspired by a recent work by J. Bellazzini and D. Ruiz, is based on a version of the Mountain Pass Theorem which makes use of the so-called “monotonicity trick” by Struwe. Possibly under stronger conditions on the interaction, we expect the existence of finite-energy traveling waves in the whole subsonic range, this is an ongoing work.