Potential well theory for the derivative nonlinear Schrödinger equation

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 $u_{0} \in H^{1} (ℝ)$ satisfies the mass condition $∥ u_{0} ∥_{L^{2}}^{2} < 4 π$ , the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation above for general $b \in ℝ$ , which corresponds exactly to the $4 π$ -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 $4 π$ -mass condition and algebraic solitons.

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Keywords

derivative nonlinear Schrödinger equation, solitons, potential wells, variational methods

Mathematical Subject Classification 2010

Primary: 35Q55, 35Q51, 37K05

Secondary: 35A15

Milestones

Received: 29 June 2019

Revised: 15 September 2019

Accepted: 21 November 2019

Published: 18 May 2021

Authors

Masayuki Hayashi
Research Institute for Mathematical Sciences Kyoto University Kyoto Japan

Vol. 14, No. 3, 2021

Masayuki Hayashi

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors