On the one dimensional cubic NLS in a critical space

Marco Bravin; Luis Vega

doi:10.3934/dcds.2021203

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2022, Volume 42, Issue 6: 2563-2584. Doi: 10.3934/dcds.2021203

This issue Previous Article Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields Next Article Higher dimensional topology and generalized Hopf bifurcations for discrete dynamical systems

On the one dimensional cubic NLS in a critical space

Marco Bravin^, and
Luis Vega

BCAM - Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Basque Country - Spain

^*Corresponding author: Marco Bravin
^*Corresponding author: Marco Bravin

Received: July 2021

Early access: January 2022

Published: June 2022

Marco Bravin is supported by ERC-2014-ADG project HADE Id. 669689 (European Research Council). Luis Vega is supported by ERC-2014-ADG project HADE Id. 669689 (European Research Council), MINECO grant BERC 2018-2021, PGC2018-094522-B-I00 and SEV-2017-0718 (Spain)

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Abstract

In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.

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Mathematics Subject Classification: Primary: 35Q55, 34A12, 37L50.

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