Segal-Wilson approach to integrable systems and Riemann-Hilbert problems

In this paper a theory is developed for obtaining families of solutions to the KdV equation by formulating a Riemann-Hilbert problem with an appropriate shift. The theory builds on the classical work of Segal and Wilson [17] in which families of solutions are indexed on closed subspaces W of a space of functions on the unit circle admitting a direct sum decomposition H = H+ circle plus H-(H+, H- are subspaces of functions holomorphic respectively inside and outside the unit disk). The theory developed in this paper lends itself easily to obtaining explicit solutions. Examples where the subspace W can be associated to soliton type solutions are considered. More complex systems where singularities and Riemann surfaces play a role are also presented. In the last section the connection of our results to the tau-function is analyzed. The theory developed in this paper can easily be applied to other integrable systems and, eventually, to discrete integrable systems. (C) 2016 Elsevier Inc. All rights reserved.