The small oscillations of solitons in 2D Bose-Einstein condensates are investigated by solving the Kadomtsev-Petviashvili equation which is valid when the velocity of the soliton approaches the speed of sound. We show that the soliton is stable and that the lowest excited states obey the same dispersion law as the one of the stable branch of excitations of a 1D gray soliton in a 2D condensate. ...

We numerically investigate transverse stability and instability of so-called cnoidal waves, i.e., periodic traveling wave solutions of the Kortewegde Vries equation, under the time-evolution of the Kadomtsev-Petviashvili equation. In particular, we find that in KP-I small amplitude cnoidal waves are stable (at least for spatially localized perturbations) and only become unstable above a certain...

The Kadomstev-Petviashvili (KP) equation is a wellknown modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation. In this paper it is shown that the KP equation is also the relevant modulation equation for bifurcation from periodic travelling waves when the wave action flux has a critical point. Moreover, the emergent KP equation arises in a...

Abstract We justify rigorously the convergence of the amplitude of solutions of Nonlinear-Schrödinger type Equations with non zero limit at infinity to an asymptotic regime governed by the Kortewegde Vries equation in dimension 1 and the Kadomtsev-Petviashvili I equation in dimensions 2 and more. We get two types of results. In the one-dimensional case, we prove directly by energy bounds that t...

This work describes a classification of the N-soliton solutions of the Kadomtsev-Petviashvili II equation in terms of chord diagrams of N chords joining pairs of 2N points. The different classes of N-solitons are enumerated by the distribution of crossings of the chords. The generating function of the chord diagrams is expressed as a continued fraction, special cases of which are moment generat...