Solitons and Geometry

Abstract

The biggest advance in the theory of nonlinear equations in recent years has been the invention of the so-called inverse scattering method by Gardner et al. [1] in 1967, who used it to solve the initial value problem for the Korteweg-de Vries [2] equation. The method was systematized by Lax [3] in 1968 and it was given definitive respectability by Zacharov and Shabat [4] in 1971 when they solved the Schröding er equation with a cubic nonlinearity. By now, a large number of equations can be solved this way [5]. This method, in the language of Lax [3], is the following: Given an evolution equation, say, of the form

$$\frac{{\partial u}}{{\partial u}} = K\left( u \right),$$

(1.1)

one finds firstly operators L and B such that Equation (1.1) is equivalent to the relation

$$\frac{{\partial L}}{{\partial t}} = i\left[ {L,B} \right],$$

(1.2)

In this case the spectrum of L does not depend on time and one says that the flow of L is isospectral. Secondly, one tries to associate a scattering problem with the operator L (which acts, for instance, on a Hilbert space). This typically restricts the type of asymptotic behavior allowed on u. Finally, the direct and inverse scattering problems associated with L must be solved.

Lectures given at NATO Advanced Study Institute on Nonlinear. Equations in Physics and Mathematics, Istanbul, August 1977.

Supported in part by the NSF Grant No. GP-40768X.