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+⊕H−H=H+⊕H− (H+,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 τ-function is analyzed. The theory developed in this paper can easily be applied to other integrable systems and, eventually, to discrete integrable systems.