Vol. 14, No. 3, 2021

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Potential well theory for the derivative nonlinear Schrödinger equation

Masayuki Hayashi

Vol. 14 (2021), No. 3, 909–944

We consider the following nonlinear Schrödinger equation of derivative type:

itu + x2u + i|u|2 xu + 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 u0 H1() 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.

derivative nonlinear Schrödinger equation, solitons, potential wells, variational methods
Mathematical Subject Classification 2010
Primary: 35Q55, 35Q51, 37K05
Secondary: 35A15
Received: 29 June 2019
Revised: 15 September 2019
Accepted: 21 November 2019
Published: 18 May 2021
Masayuki Hayashi
Research Institute for Mathematical Sciences
Kyoto University