Local and global well-posedness for the kinetic derivative NLS on $\mathbb{R}$

TL;DR

This paper studies the well-posedness of the kinetic derivative nonlinear Schrödinger equation on the real line, establishing local results via energy methods and global results in the dissipative case.

Contribution

It provides the first well-posedness results for KDNLS on $\

Findings

Local well-posedness in $H^2$ and $H^{2}igcap H^{1,1}$ spaces.

Global well-posedness for the dissipative case with $eta<0$.

Use of gauge transformations to handle resonant interactions.

Abstract

We investigate the local and global well-posedness of the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on $\mathbb{R}$, described by \[ i\partial_t u + \partial_x^2 u = i\alpha \partial_x (|u|^2 u) + i\beta \partial_x (H(|u|^2) u), \] where $\alpha, \beta \in \mathbb{R}$, and $H$ represents the Hilbert transformation. For KDNLS, the $L^2$ norm of a solution is decreasing (resp. increasing, conserved) when $\beta$ is negative (resp. positive, zero). Focusing on the Sobolev spaces $H^2$ and $H^2 \cap H^{1,1}$, we establish local well-posedness via the energy method combined with gauge transformations to address resonant interactions in both cases of negative and positive $\beta$. For the dissipative case $\beta < 0$, we further demonstrate global well-posedness by deriving an a priori bound in $H^2$.