During the 90's a perturbation theory was developed for the resolvents in which the perturbation was allowed to be large in norm but small in rank. Given a linear constant coefficient ordinary differential equation, its solution oper- ator depends strongly on the form of the boundary conditions. However, modifying boundary conditions results as low rank perturbation of the solution operator and thus the low rank perturbation theory is available. A fundamental result is that the growth of the resolvent as a meromorphic operator-valued function is robust under low rank perturbations, even so the place and nature of singularities can change dramatically. The evolution equations governing solitons often contain differential operator (in space variable) of third order. The problem of representing the solutions of third order linear equations arises and is for general boundary conditions on finite intervals highly nontrivial.
The interest in writing down the exact solutions for such equations comes form
the success of such 'nonlinear Fourier-transform' method on unbounded intervals.
A.S Fokas has written several articles on this.