Local well-posedness for the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

2025/05/27 by Luc Molinet, Tomoyuki Tanaka, Molinet, Luc +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2505.20883

openalex publication_date 2025/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the derivative nonlinear Schrödinger equation on the real line, with a background function ψ(t,x)∈ L^∞(ℝ²) 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^∞ function is unconditionally locally well-posed in H^s(ℝ) for s>3/4 . As a byproduct, we also establish local well-posedness in the Zhidkov space.

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