Multiple branches of travelling waves for the Gross Pitaevskii equation

Explicit solitary waves are known to exist for the Kadomtsev-Petviashvili-I (KP-I) equation in dimension 2. We first address numerically the question of their Morse index. The results confirm that the lump solitary wave has Morse index one and that the other explicit solutions correspond to excited states. We then turn to the 2D Gross-Pitaevskii (GP) equation which in some long wave regime converges to the (KP-I) equation. Numerical simulations already showed that a branch of travelling waves of (GP) converges to a ground state of (KP-I), expected to be the lump. In this work, we perform numerical simulations showing that the other explicit solitary waves solutions to the (KP-I) equation give rise to new branches of travelling waves of (GP) corresponding to excited states. 1991 Mathematics Subject Classification. 35B38, 35C07, 35J61, 35Q40, 35Q55.