Nonlinear Fourier transforms and the mKdV equation in the quarter plane

TL;DR

This paper rigorously applies the Fokas unified transform method to the mKdV equation in the quarter plane, providing detailed estimates for nonlinear Fourier transforms and constructing smooth solutions under limited regularity.

Contribution

It offers a rigorous implementation of the unified transform method for the mKdV equation in the quarter plane, including detailed estimates and solution construction under weaker assumptions.

Findings

Provided detailed estimates for nonlinear Fourier transforms.

Constructed solutions that are $C^1$ in time and $C^3$ in space.

Extended the method to cases with limited regularity and decay.

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.