Mathematics

Between 1967 and 1974, Kruskal, Miura, Green and Gardner published a series of papers where they proved the Integrability of a Nonlinear PDE, called the Korteweg - de Vries equation (KdV), and find a new metod to find solution of it for a given initial datum $u_0(x)$, the "Inverse Scattering Method" (ISM). In this seminar, we apply the ISM to the problem of KdV with periodic initial condition, i.e. $u_0(x +T)=u_0(x)$. We show that this problem is linked to the algebraic geometry problem of inverting Abelian Integrals on a compact Riemann Surface ( also called "Jacobi inversion problem").

This seminar is based on the works of B. Dubrovin, S. Novikov, A. Its and V. Matveev.