On the derivative nonlinear Schr{\"o}dinger equation on the half line with Robin boundary condition

TL;DR

This paper studies the derivative nonlinear Schrödinger equation on the half-line with Robin boundary conditions, demonstrating conditions for solution blow-up and analyzing the stability of standing waves using variational methods.

Contribution

It provides new results on blow-up solutions and stability analysis for the derivative nonlinear Schrödinger equation with Robin boundary conditions.

Findings

Existence of blow-up solutions under certain conditions.

Stability and instability results for standing waves.

Application of virial and variational techniques.

Abstract

We consider the Schr\"odinger equation with nonlinear derivative term on $[0,+\infty)$ under Robin boundary condition at $0$. Using a virial argument, we obtain the existence of blowing up solutions and using variational techniques, we obtain stability and instability by blow up results for standing waves.