Inverse problems related to integrable nonlinear partial differential equations

Abstract

The so-called inverse scattering method manages solutions of integrable nonlinear partial differential equation as the Fourier method does for linear p.d.e. In its most primitive form, several different inverse problems were involved. It has been shown recently that all the inverse problems of IST can be put in a common structure, that of a unique integral equation for a matrix valued function ψ which yields the solution ψ1 of the n.l.p.d.e and the constraints, if any, which restrict the class of solutions one can obtain. The various integrable n.l.p.d.e are related to the dimension of space and that of matrices. This result enables a classification of integrable n.l.p.d.e. It also clearly shows that the only remaining difficult point is to associate classes of solutions of n.l.p.d.e to classes of kernels of the integral equation. Although this problem here is almost virgin, it is certainly not new in the study of Inverse Problems, where overlooking it is the classical source of illposedness.