Nash--Moser iteration and singular perturbations

2009/07/27 by Benjamin Texier, Texier, Benjamin, Kevin Zumbrun +1
Mathematics · Physics and Astronomy · #35B35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Numerical methods for differential equations #math.AP #msc:35B35

paper · pdf · doi:10.48550/arxiv.0907.4661

Final version; general presentation completely revised; new sample application to systems of quasilinear Schrödinger equations

openalex publication_date 2009/07/27 · arxiv created 2011/07/07 · arxiv updated 2011/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a simple and easy-to-use Nash--Moser iteration theorem tailored for singular perturbation problems admitting a formal asymptotic expansion or other family of approximate solutions depending on a parameter \eps→ 0. The novel feature is to allow loss of powers of \eps as well as the usual loss of derivatives in the solution operator for the associated linearized problem. We indicate the utility of this theorem by describing sample applications to (i) systems of quasilinear Schrödinger equations, and (ii) existence of small-amplitude profiles of quasilinear relaxation systems.

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