Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane. Issue 1 (January 2016)

• Record Type: Journal Article
• Authors: Lenells, Jonatan
• Is Part Of: Studies in applied mathematics. Volume 136:Issue 1(2016)
• Page Start: 3
• Page End: 63
• Publication Date: 2016-01
• Subjects: Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12089
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract : The unified transform method introduced by Fokas can be used to analyze initial‐boundary value problems for integrable evolution equations. The method involves several steps, including the definition of spectral functions via nonlinear Fourier transforms and the formulation of a Riemann‐Hilbert problem. We provide a rigorous implementation of these steps in the case of the mKdV equation in the quarter plane under limited regularity and decay assumptions. We give detailed estimates for the relevant nonlinear Fourier transforms. Using the theory of L 2 ‐RH problems, we consider the construction of quarter plane solutions which are C 1 in time and C 3 in space.