Potential well theory for the derivative nonlinear Schr\"{o}dinger equation

We consider the following nonlinear Schr\"{o}dinger 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 Schr\"{o}dinger 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 $\| u_0\|_{L^2}^2 <4\pi$, 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.Comment: To 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

Stability of algebraic solitons for nonlinear Schr\"{o}dinger equations of derivative type: variational approach

We consider the following nonlinear Schr\"{o}dinger 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 a gauge equivalent form of well-known derivative nonlinear Schr\"{o}dinger (DNLS) equation. The equation can be considered as a generalized equation of DNLS while preserving both $L^2$-criticality and Hamiltonian structure. If $b>-3/16$, the equation has algebraically decaying solitons, which we call algebraic solitons, as well as exponentially decaying solitons. In this paper we study stability properties of the solitons by variational approach and prove that if $b<0$, all solitons including algebraic solitons are stable in the energy space. The stability of algebraic solitons gives the counterpart of the previous instability result for the case $b>0$.Comment: 24 pages, 1 figur

The Cauchy problem for the logarithmic Schr\"odinger equation revisited

We revisit the Cauchy problem for the logarithmic Schr\"odinger equation and construct strong solutions in $H^1$, the energy space, and the $H^2$-energy space. The solutions are provided in a constructive way, which does not rely on compactness arguments, that a sequence of approximate solutions forms a Cauchy sequence in a complete function space and then actual convergence is shown to be in a strong sense.Comment: 30 page

Low regularity solutions to the logarithmic Schrodinger equation

We consider the logarithmic Schr{\"o}dinger equation, in various geometric settings. We show that the flow map can be uniquely extended from H^1 to L^2 , and that this extension is Lipschitz continuous. Moreover, we prove the regularity of the flow map in intermediate Sobolev spaces.Comment: Some typos fixed. A flaw corrected in Section