Y. Martel, F. Merle, P. Raphael, J. Szeftel, “Near soliton dynamics and singularity formation for <nobr>$L^2$</nobr> critical problems”, Uspekhi Mat. Nauk, 2014, Volume 69, Issue 2(416),Pages <nobr>77

This article is cited in 6 papers

Near soliton dynamics and singularity formation for $L^2$ critical problems

Y. Martel^a, F. Merle^bc, P. Raphael^de, J. Szeftel^fg

^a Ècole Polytechnique, Centre de Mathématiques, Palaiseau, France
^b Institut des Hautes Études Scientifiques, Bures-sur-Ivette, France
^c Université de Cergy-Pontoise, Cergy-Pontoise, France
^d Université Paul Sabatier, Toulouse
^e Institut Universitaire de France, Paris
^f Centre National de la Recherche Scientifique, Paris, France
^g Ècole Normale Supérieure, Paris, France

Abstract: This survey reviews the state of the art concerning singularity formation for two canonical dispersive problems: the $L^2$ critical non-linear Schrödinger equation and the $L^2$ critical generalized KdV equation. In particular, the currently very topical question of classifying flows with initial data near a soliton is addressed.
Bibliography: 72 titles.

Keywords: non-linear Schrödinger equation, critical equation, generalized Korteweg–de Vries equation, blowup, soliton, blowup profile, qualitative behaviour of solutions, non-linear dispersive equation.

UDC: 517.95

MSC: 35Q55, 35Q53, 35B40, 35B44, 35B35

Received: 27.10.2013

DOI: 10.4213/rm9574