Numerical computation of the finite-genus solutions of the Korteweg–de Vries equation via Riemann–Hilbert problems

In this letter we describe how to compute the finite-genus solutions of the Korteweg–de Vries equation using a Riemann–Hilbert problem that is satisfied by the Baker–Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the Baker–Akhiezer function. This method has the benefit that the space and time dependence of the Baker–Akhiezer function appear in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann–Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all finite-genus solutions of the KdV equation.