A Riemann–Hilbert problem for the finite-genus solutions of the KdV equation and its numerical solution

We derive a Riemann–Hilbert problem satisfied by the Baker–Akhiezer function for the finite-gap solutions of the Korteweg–de Vries (KdV) equation. As usual for Riemann–Hilbert problems associated with solutions of integrable equations, this formulation has the benefit that the space and time dependence appears 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 periodic and quasi-periodic finite-genus solutions of the KdV equation.