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Annales scientifiques de l'ENS - Parutions - série 4, 51 (2018)

Parutions < série 4, 51

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 51, fascicule 3 (2018)

Yvan Martel, Pierre Raphaël
Strongly interacting blow up bubbles for the mass critical nonlinear Schrödinger equation
Annales scientifiques de l'ENS 51, fascicule 3 (2018), 701-737

Télécharger cet article : Fichier PDF

Résumé :
Bulles d'explosion en interaction forte pour l'équation de Schrödinger non linéaire critique pour la masse
On considère l'équation de Schrödinger non linéaire critique pour la masse en dimension deux i_tu+u+|u|^2u=0, tR ,xR ^2.(SNL) Soit Q la solution positive et état fondamental de l'équation Q-Q+Q^3=0. On construit une nouvelle classe d'ondes solitaires multiples basées sur Q : étant donné un entier K2, il existe une solution globale (pour t>0) u(t) de (SNL) qui se décompose asymptotiquement en une somme d'ondes solitaires centrées sur les sommets d'un polygone régulier et qui se concentrent à un taux logarithmique quand t+, de sorte que la solution explose en temps infini u(t)_L^2|t|quandt+. Comme conséquence de la symétrie pseudo-conforme du flot de (SNL), on obtient le premier exemple d'une solution v(t) de (SNL) qui explose en temps fini avec un taux strictement supérieur au taux pseudo-conforme v(t)_L^2| |t|t|quandt0. Cette solution concentre K bulles en un point x_0^2, c'est-à-dire |v(t)|^2KQ_L^2^2_x_0 quand t0. Ces comportements particuliers sont dus aux interactions fortes entre les ondes solitaires, par opposition avec les résultats précédents sur les ondes solitaires multiples pour (SNL) où les interactions n'affectent pas le comportement global des ondes.

Mots-clefs : Équation de Schrödinger non linéaire, non-linéarité critique, explosion, multi-solitons.

Abstract:
We consider the mass critical two dimensional nonlinear Schrödinger equation i_tu+u+|u|^2u=0, tR ,xR ^2.(NLS) Let Q denote the positive ground state solution of Q-Q+Q^3=0. We construct a new class of multi-solitary wave solutions of (NLS) based on Q: given any integer K2, there exists a global (for t>0) solution u(t) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K-sided regular polygon and concentrating at a logarithmic rate as t+, so that the solution blows up in infinite time with the rate u(t)_L^2|t|ast+. Using the pseudo-conformal symmetry of the (NLS) flow, this yields the first example of solution v(t) of (NLS) blowing up in finite time with a rate strictly above the pseudo-conformal one, namely, v(t)_L^2| |t|t|ast0. Such a solution concentrates K bubbles at a point x_0^2, that is |v(t)|^2KQ_L^2^2_x_0 as t0. These special behaviors are due to strong interactions between the waves, in contrast with previous works on multi-solitary waves of (NLS) where interactions do not affect the global behavior of the waves.

Keywords: Nonlinear Schrödinger equation, critical nonlinearity, blow up, multi-solitons

Class. math. : 35Q55; 35B44, 37K40.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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