Local well-posedness for the derivative nonlinear Schr\"odinger equation with nonvanishing boundary conditions

TL;DR

This paper proves local well-posedness for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions, including non-decaying solutions like dark solitons, using an energy method with correction terms.

Contribution

It introduces a novel energy method with correction terms to establish local well-posedness for perturbations around non-decaying backgrounds in the derivative NLS.

Findings

Unconditional local well-posedness in H^s for s > 3/4

Well-posedness in Zhidkov space

Applicability to non-decaying solutions like dark solitons

Abstract

We consider the derivative nonlinear Schr\"odinger equation on the real line, with a background function $\psi(t,x)\in L^\infty(\mathbb{R}^2)$ that satisfies suitable conditions. Such a function may, for example, be a non-decaying solution of the equation, such as a dark soliton. By developing the energy method with correction terms, we prove that the Cauchy problem for perturbations around such an $L^\infty$ function is unconditionally locally well-posed in $ H^s(\mathbb{R}) $ for $ s>3/4 $. As a byproduct, we also establish local well-posedness in the Zhidkov space.