A Riemann–Hilbert method to algebro-geometric solutions of the Korteweg–de Vries equation

Abstract

A proof without using Dubrovin’s equations is given for the Its–Matveev formula for algebro-geometric solutions of the Korteweg–de Vries (KdV) equation. It is shown that the Baker–Akhiezer (BA) function of the KdV equation can be described in terms of two types of holomorphic vector Riemann–Hilbert problems with σ1-symmetry condition in the complex k-plane. Further, a sufficient condition on determining whether the algebro-geometric KdV solution has mKdV-relevant property is presented. This provides an explanation for the classical Miura transformation in terms of the deformations of associated algebro-geometric datum. Our main tools include algebraic curve and Riemann surface, Riemann theta function, Riemann–Hilbert problem and associated deformation procedure.