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.

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

Keywords

Mathematical Subject Classification 2010

Milestones

Authors