#### 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
##### Abstract

We consider the following nonlinear Schrödinger equation of derivative type:

$i\phantom{\rule{0.3em}{0ex}}{\partial }_{t}u+{\partial }_{x}^{2}u+i|u{|}^{2}\phantom{\rule{0.3em}{0ex}}{\partial }_{x}u+b|u{|}^{4}u=0,\phantom{\rule{1em}{0ex}}\left(t,x\right)\in ℝ×ℝ,\phantom{\rule{0.3em}{0ex}}b\in ℝ.$

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 ${u}_{0}\in {H}^{1}\left(ℝ\right)$ satisfies the mass condition $\parallel {u}_{0}{\parallel }_{{L}^{2}}^{2}<4\pi$, 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 $4\pi$-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 $4\pi$-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