Vol. 10, No. 5, 2017

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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

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 u0 is global if u0L22 < 4π 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 u0 satisfies u0L22 = 4π and the momentum P(u0) is negative.

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variational structure, generalized derivative nonlinear Schr&ouml;dinger equation, global existence
Mathematical Subject Classification 2010
Primary: 35Q55
Received: 8 October 2016
Revised: 28 February 2017
Accepted: 3 April 2017
Published: 1 July 2017
Noriyoshi Fukaya
Department of Mathematics
Graduate School of Science
Tokyo University of Science
Shinjuku, Tokyo 162-8601
Masayuki Hayashi
Department of Applied Physics
Waseda University
Shinjuku, Tokyo 169-8555
Takahisa Inui
Department of Mathematics
Graduate School of Science
Kyoto University
Kyoto City, Kyoto 606-8502