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Nonlinear Fourier Transforms and the mKdV Equation in the Quarter Plane
Author(s) -
Lenells Jonatan
Publication year - 2016
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12089
Subject(s) - mathematics , nonlinear system , mathematical analysis , fourier transform , integrable system , plane (geometry) , hilbert space , split step method , quarter (canadian coin) , boundary value problem , complex plane , partial differential equation , geometry , physics , archaeology , quantum mechanics , history
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.