On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schroedinger equations

2000/06/26 by Axel Gruenrock, Gruenrock, Axel
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations #advanced mathematical theories #math.AP

paper · pdf · doi:10.48550/arxiv.math/0006195

arxiv created 2000/06/26 · openalex publication_date 2000/06/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Cauchy- and periodic boundary value problem for the nonlinear Schroedinger equations in n space dimensions [u_t - iΔu = (∇ u)^β, |β|=m ≥ 2, u(0)=u₀ ∈ H^s+1_x] is shown to be locally well posed for s > s_c := (n)/(2) - (1)/(m-1), s ≥ 0. In the special case of space dimension n=1 a global L²-result is obtained for NLS with the nonlinearity N(u)= ∂_x (u ²). The proof uses the Fourier restriction norm method.

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