Almost global existence for cubic nonlinear Schrödinger equations in one space dimension

Jason Murphy; Fabio Pusateri

doi:10.3934/dcds.2017089

Article Contents

2017, Volume 37, Issue 4: 2077-2102. Doi: 10.3934/dcds.2017089

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Almost global existence for cubic nonlinear Schrödinger equations in one space dimension

Jason Murphy^1, , and
Fabio Pusateri^2,

1.
Department of Mathematics, University of California, 970 Evans Hall, Berkeley, CA 94720-3840, USA
2.
Department of Mathematics, Princeton University, Fine Hall, 304 Washington Rd, Princeton, NJ 08544, USA

Author Bio: E-mail address: murphy@math.berkeley.edu; E-mail address: fabiop@math.princeton.edu
* Corresponding author: Jason Murphy
* Corresponding author: Jason Murphy

Received: April 30, 2016

Revised: October 31, 2016

Published: May 2017

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Abstract

We consider non-gauge-invariant cubic nonlinear Schrödinger equations in one space dimension.We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^∞$ decay up to time $\exp(C\varepsilon^{-2})$. We also exhibit norm growth beyond this time for a specific choice of nonlinearity.

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Mathematics Subject Classification: Primary:35Q55.

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