Rarefaction pulses for the Nonlinear Schrödinger Equation in the transonic limit.

David Chiron (1) , Mihai Maris (2)

1 JAD - Laboratoire Jean Alexandre Dieudonné
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We investigate the properties of finite energy travelling waves to the nonlinear Schrödinger equation with nonzero conditions at infinity for a wide class of nonlinearities. In space dimension two and three we prove that travelling waves converge in the transonic limit (up to rescaling) to ground states of the Kadomtsev-Petviashvili equation. Our results generalize an earlier result of F. Béthuel, P. Gravejat and J-C. Saut for the two-dimensional Gross-Pitaevskii equation, and provide a rigorous proof to a conjecture by C. Jones and P. H. Roberts about the existence of an upper branch of travelling waves in dimension three.

Domaines: Equations aux dérivées partielles [math.AP]

David Chiron, Mihai Maris. Rarefaction pulses for the Nonlinear Schrödinger Equation in the transonic limit.. Communications in Mathematical Physics, 2014, 326 (2), pp.329-392. ⟨hal-00874591⟩