On the Dirac delta as initial condition for nonlinear Schrödinger equations

Abstract

In this article we will study the initial value problem for some Schrödinger equations with Diraclike initial data and therefore with infinite L 2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schrödinger maps through the so called Hasimoto transformation. Résumé Dans cet article onétudie le problème de Cauchy pour deséquations de Schrödinger avec donnée initiale de type Dirac et donc avec masse L 2 infinie, obtenant des résultats positifs pour les non linéarités sous-critiques. Dans le cas critique et en dimension un, on montre qu'après une certaine renormalisation la solution correspondante est d'énergie finie. On en déduit un résultat de stabilité dans le cas défocalisant. Ces problèmes sont liésà l'existence d'une dynamique singulière des applications de type Schrödinger par la transformation de Hasimoto. -----------------------------------------1

Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation

“…L. Vega is supported by an ERCEA Advanced Grant 2014 669689 -HADE, by the MEIC project MTM2014-53850-P and MEIC Severo Ochoa excellence accreditation SEV-2013-0323. 1 scattering where self-similar solutions naturally appear: we refer to Hayashi and Naumkin [15,14], which was revisited by Germain, Pusateri and Rousset [10] and Harrop-Griffiths [12]. Self-similar solutions and the (mKdV) flow are also relevant as a model for the behavior of vortex filament in fluid dynamics.…”

“…Also, any solution to (1) which tends to 0 as y Ñ`8 is one of the V κ . If furthermore κ P p´1, 1q, V κ is defined on and…”

“…Theorem 1. Given ǫ P t˘1u, k P`1 2 , 4 7˘a nd c, α P with c 2`α2 ă ε 0 small enough, there exist A " Apc, αq and a real valued function V P 1 p q solution to (1) such that @ξ ą 0, e´i ξ 3V pξq " χpξqe ia ln |ξ|˜A`B e 2ia ln |ξ| e´i…”

The Cauchy problem for the Landau–Lifshitz–Gilbert equation in BMO and self-similar solutions

“…there exists a unique solution m ∈ X(R N × R + ; S 2 ) of (LLG α ) with initial condition m 0 such that 3 We refer the reader to Theorem A.5 in the Appendix and to [17] for precise statements of these results. 4 See footnote in Section 3.3 for the definition of the Morrey space M 2,2 (R N ).…”

Обращение Времени Для Модифицированных Осцилляторов

“…работы [32], [39] по поводу некорректности некоторых канонических дисперсионных уравнений). Второй случай, отвечающий (2.3), рассматривается аналогично.…”