On the initial value problem for a class of nonlinear integral evolution equations including the sine–Hilbert equation

Abstract

A method for solving a class of nonlinear singular integral evolution equations for decaying initial values on the line is presented. The underlying scattering problem is a matrix Riemann–Hilbert problem. Scattering analysis shows that the spectrum is purely discrete. An application is to the so-called sine–Hilbert equation Hθt =−c sin θ, where c is a constant and H denotes the Hilbert transform.