Potential well theory for the derivative nonlinear Schrödinger equation

Abstract

We consider the following nonlinear Schrödinger equation of derivative type: \begin{equation}i \partial_t u + \partial_x^2 u +i |u|^{2} \partial_x u +b|u|^4u=0 , \quad (t,x) \in \mathbb{R}\times\mathbb{R}, \ b \in\mathbb{R}. \end{equation} If $b=0$, this equation is known as a gauge equivalent form of well-known derivative nonlinear Schrödinger equation (DNLS), which is mass critical and completely integrable. The equation 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 $u_0\in H^1(\mathbb{R})$ satisfies the mass condition $\Vert u_0{\Vert }_{L^2}^2<4\pi$, the corresponding solution is global and bounded. In this paper we first establish the mass condition on the equation for general $b\in \mathbb{R}$, which is exactly corresponding to $4\pi$-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 $4\pi$-mass condition and algebraic solitons.

arXiv commentTo appear in Analysis & PDE. This paper was submitted to the journal on June 29, 2019. The author cited the revised version of the paper by Kwon and Wu (see arXiv:1603.03745) and removed Appendix A

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Cited by in corpus (4)

• Traveling waves for a nonlinear Schrödinger system with quadratic interaction
• Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach
• The stability of degenerate solitons for derivative nonlinear Schrodinger equations
• Instability of degenerate solitons for nonlinear Schrödinger equations with derivative