Nonlinear Evolution Equations from an Inverse Spectral Problem

It is known that a number of nonlinear evolution equations can be related to an inverse spectral problem and, then, (in principle) linearized by solving a linear integral equation. In this lecture the opposite way is followed, i.e. one starts from a linear inverse spectral problem and deduces the possible nonlinear evolution equations which can be solved by using it. In order to fix the most general possible time evolution and space dependence of the spectral data in the considered spectral problem only consistency requirements are used. Then, a related nonlinear evolution equation for a matrix field is derived. In general, it results that this matrix field satisfies some additional non linear differential constraints. The main advantage of the method is that it allows a systematic search of all possible nonlinear evolution equations of given order which can be solved by using the considered inverse spectral problem. New classes of first and second order evolution equations are obtaine..