Local well-posedness for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions
Luc Molinet, Tomoyuki Tanaka
TL;DR
This work addresses local well-posedness for the derivative nonlinear Schrödinger equation on ℝ with nonvanishing boundary data. By decomposing the solution into a prescribed background ψ and a perturbation u, and by employing an energy method with correction terms together with refined Strichartz estimates, the authors obtain unconditional local well-posedness in H^s(ℝ) for s>3/4 when λ=2μ or λ=0, and for s≥1 in the generic case, with a Zhidkov-space corollary. A central challenge is derivative loss from the derivative nonlinearity and the presence of a non-L^2 background, which is overcome via a modified energy that cancels the worst resonant interactions and a low-frequency weight to handle the background. The results extend the well-posedness theory to nonvanishing boundary conditions and provide robust tools (modified energies, frequency envelopes) that relate to the equation’s resonant structure and background nonlinearity. The paper thus advances understanding of DNLS in non-L^2 settings and yields precise regularity thresholds for well-posedness with potential implications for long-time behavior and stability of dark-soliton-type solutions.
Abstract
We consider the derivative nonlinear Schrödinger equation on the real line, with a background function $ψ(t,x)\in L^\infty(\mathbb{R}^2)$ 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.