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

Using the Unified Transform, also known as the Fokas method, 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.