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2021 Potential well theory for the derivative nonlinear Schrödinger equation
Masayuki Hayashi
Anal. PDE 14(3): 909-944 (2021). DOI: 10.2140/apde.2021.14.909

Abstract

We consider the following nonlinear Schrödinger equation of derivative type:

itu+x2u+i|u|2xu+b|u|4u=0,(t,x)×,b.

If b=0, this equation is known as a standard derivative nonlinear Schrödinger equation(DNLS), which is mass-critical and completely integrable. The equation above can be considered as a generalized equation of DNLS while preserving mass-criticality and Hamiltonian structure. For DNLS it is known that if the initial data u0H1() satisfies the mass condition u0L22<4π, the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation above for general b, which corresponds exactly to the 4π-mass condition for DNLS, and then characterize it from the viewpoint of potential well theory. We see that the mass-threshold value gives the turning point in the structure of potential wells generated by solitons. In particular, our results for DNLS give a characterization of both the 4π-mass condition and algebraic solitons.

Citation

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Masayuki Hayashi. "Potential well theory for the derivative nonlinear Schrödinger equation." Anal. PDE 14 (3) 909 - 944, 2021. https://doi.org/10.2140/apde.2021.14.909

Information

Received: 29 June 2019; Revised: 15 September 2019; Accepted: 21 November 2019; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/apde.2021.14.909

Subjects:
Primary: 35Q51, 35Q55, 37K05
Secondary: 35A15

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.14 • No. 3 • 2021
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