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

Luc Molinet; Tomoyuki Tanaka

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arXiv · 2505.20883

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

Luc Molinet·Tomoyuki Tanaka

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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.

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Luc Molinet, Tomoyuki Tanaka. 2025-05-27. Local well-posedness for the derivative nonlinear Schr\"odinger equation with nonvanishing boundary conditions. https://arxiv.org/abs/2505.20883

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