Nonlinear Schrödinger and korteweg-de Vries are reductions of self-dual Yang-Mills

Abstract

The nonlinear Schrödinger (NS) and KdV equations are shown to be reductions of the self-dual Yang-Mills (SDYM) equations. A correspondence between solutions of the NS and KdV equations and certain holomorphic vector bundles on a complex line bundle over the Riemann sphere is derived from Ward's SDYM twistor correspondence. Remarkably the twistor correspondence generalizes to the NS and KdV hierarchies when complex line bundles of higher Chern class are used. We discuss solitons and inverse scattering.