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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 wavesdoi:10.1088/1361-6544/aab4cc fatcat:r7axf2ygjfhxpi4pazkpr7xfu4