Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

TL;DR

This paper studies the nonlinear Fourier transforms related to the sine-Gordon equation in the quarter plane, providing foundational analysis for solving the equation via Riemann-Hilbert problems and long-time asymptotics.

Contribution

It offers an extensive analysis of the nonlinear Fourier transforms and eigenfunctions under weak regularity assumptions, advancing the mathematical framework for sine-Gordon solutions.

Findings

Analysis of spectral functions $a,b,A,B$ under weak regularity

Establishment of properties of eigenfunctions for sine-Gordon

Framework for long-time asymptotic analysis using steepest descent

Abstract

The solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann-Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.