The nonlinear Schrödinger equation on the half-space

2024/11/25 by A. Alexandrou Himonas, Himonas, A. Alexandrou, Fangchi Yan +1 · 3 citations
Mathematics · Physics and Astronomy · #35G16 #35G31 #35Q55 #37K10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.2411.16610

openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work studies the initial-boundary value problem for both the linear Schrödinger equation and the cubic nonlinear Schrödinger equation on the half-space in higher dimensions (n≥ 2). First, the forced linear problem is solved on the half-space via the Fokas method and then using the obtained solution formula new and interesting linear estimates are derived with data and forcing in appropriate spaces. Second, the well-posedness of the nonlinear problem on the half-space is proved with initial data in Sobolev spaces H^s(ℝⁿ₊), with s>(n)/(2)-1, and boundary data in natural Bourgain spaces B^s that reflect the boundary regularity of the linear problem. The proof method consists of showing that the iteration map defined via the Fokas solution formula is a contraction by establishing sharper trilinear estimates. The presence of the boundary introduces solution spaces that involve temporal Bourgain spaces.

Cited by

• The periodic Camassa-Holm equation by the Riemann-Hilbert problem approach
• Mathematical results for the nonlinear Winter's model
• Fokas method for linear convection-diffusion equation with time-dependent coefficients and its extension to other evolution equations

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