38 research outputs found
Asymptotics of the solitary waves for the generalised Kadomtsev-Petviashvili equations
We investigate the asymptotic behaviour of the localised solitary waves for
the generalised Kadomtsev-Petviashvili equations. In particular, we compute
their first order asymptotics in any dimension
Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation
We construct families of smooth travelling-wave solutions to the inviscid
surface quasi-geostrophic equation (SQG). These solutions can be viewed as the
equivalents for this equation of the vortex anti-vortex pairs in the context of
the incompressible Euler equation. Our argument relies on the stream function
formulation and eventually amounts to solving a fractional nonlinear elliptic
equation by variational methods
Existence and properties of travelling waves for the Gross-Pitaevskii equation
This paper presents recent results concerning the existence and qualitative
properties of travelling wave solutions to the Gross-Pitaevskii equation posed
on the whole space R^N. Unlike the defocusing nonlinear Schr\"odinger equations
with null condition at infinity, the presence of non-zero conditions at
infinity yields a rather rich and delicate dynamics. We focus on the case N = 2
and N = 3, and also briefly review some classical results on the
one-dimensional case. The works we survey provide rigorous justifications to
the impressive series of results which Jones, Putterman and Roberts established
by formal and numerical arguments
On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I
The fact that the Korteweg-de-Vries equation offers a good approximation of
long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii
equation was derived several years ago in the physical literature. In this
paper, we provide a rigorous proof of this fact, and compute a precise estimate
for the error term. Our proof relies on the integrability of both the
equations. In particular, we give a relation between the invariants of the two
equations, which, we hope, is of independent interest.Comment: Final version accepted for publication in International Mathematics
Research Notices with a few minor corrections and added remark
Stabilité des solitons de l'équation de Landau-Lifshitz à anisotropie planaire
Séminaire Laurent Schwartz - EDP et applicationsCet exposé présente plusieurs résultats récents quant à la stabilité des solitons sombres de l'équation de Landau-Lifshitz à anisotropie planaire, en particulier, quant à la stabilité orbitale des trains (bien préparés) de solitons gris et à la stabilité asymptotique de ces mêmes solitons
Stability in the energy space for chains of solitons of the Landau-Lifshitz equation
International audienceWe prove the orbital stability of sums of solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy, under the assumptions that the (non-zero) speeds of the solitons are different, and that their initial positions are sufficiently separated and ordered according to their speeds