Solution of the Sturm-Liouville and the Korteweg-de-Vries equations with periodic and quasi-periodic parameters using theory of vessels

We prove the existence of solutions to the Sturm-Liouville (SL) equation -y"(x)+q(x)y(x) = s^2 y(x) with periodic and quasi-periodic potential q(x) using theory of SL vessels, implementing a Backlund transformation of SL equation. In this paper quasi-periodic means a finite sum of periodic integrable functions. The solutions for a general s are explicitly constructed in terms of the solutions zn(x), satisfying the SL equation with initial conditions zn(0)=0, zn'(0)=1 for a discrete Levinson set of numbers s=sn, n-natural number. The tau function tau(x) of the corresponding vessel realizes the given potential via the formula q(x)= - 2(ln(tau(x)))". We also prove an analogue of the inverse scattering theorem in this setting too. Using the notion of "KdV evolutionary vessel", we construct a solution of the Korteweg-de-Vries (KdV) equation q'_t = - 3/2 q q'_x + 1/4 q"'_{xxx}, which coincides for t=0 with a given (periodic or quasi-periodic) potential.