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.