A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation

Fukaya, Noriyoshi; Hayashi, Masayuki; Inui, Takahisa

doi:10.2140/apde.2017.10.1149

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A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation

Noriyoshi Fukaya, Masayuki Hayashi and Takahisa Inui

Vol. 10 (2017), No. 5, 1149–1167

DOI: 10.2140/apde.2017.10.1149

Abstract

We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schrödinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schrödinger equation (DNLS). For (DNLS), Wu (2015) proved that the solution with the initial data $u_{0}$ is global if $∥ u_{0} ∥_{L^{2}}^{2} < 4 π$ by the sharp Gagliardo–Nirenberg inequality. The variational argument gives us another proof of the global existence for (DNLS). Moreover, by the variational argument, we can show that the solution to (DNLS) is global if the initial data $u_{0}$ satisfies $∥ u_{0} ∥_{L^{2}}^{2} = 4 π$ and the momentum $P (u_{0})$ is negative.

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Keywords

variational structure, generalized derivative nonlinear Schrödinger equation, global existence

Mathematical Subject Classification 2010

Primary: 35Q55

Milestones

Received: 8 October 2016

Revised: 28 February 2017

Accepted: 3 April 2017

Published: 1 July 2017

Authors

Noriyoshi Fukaya
Department of Mathematics Graduate School of Science Tokyo University of Science Shinjuku, Tokyo 162-8601 Japan

Masayuki Hayashi
Department of Applied Physics Waseda University Shinjuku, Tokyo 169-8555 Japan

Takahisa Inui
Department of Mathematics Graduate School of Science Kyoto University Kyoto City, Kyoto 606-8502 Japan

Vol. 10, No. 5, 2017