Solution of the Korteweg-de Vries equation on the line with analytic initial potential

We present a theory of Sturm-Liouville non-symmetric vessels, realizing an inverse scattering theory for the Sturm-Liouville operator with analytic potentials on the line. This construction is equivalent to the construction of a matrix spectral measure for the Sturm-Liouville operator, defined with an analytic potential on the line. Evolving such vessels we generate Korteweg-de Vries (KdV) vessels, realizing solutions of the KdV equation. As a consequence, we prove the theorem as follows: Suppose that q(x) is an analytic function on $\mathbb {R}$R. Then there exists a closed subset $\Omega \subseteq \mathbb {R}^2$⊆R² and a KdV vessel, defined on ω. For each $x\in \mathbb {R}$xεR one can find T^x > 0 such that {x} × [ -T^x, T^x]⊆. The potential q(x) is realized by the vessel for t = 0. Since we also show that if q(x, t) is a solution of the KdV equation on $\mathbb {R}\times [0,t_0)$R×[0,t0), then there exists a vessel, realizing it, the theory of vessels becomes a universal tool to study this problem. Finally, we notice that the idea of the proof applies to a similar existence of a solution for evolutionary nonlinear Schrödinger and Boussinesq equations, since both of these equations possess vessel constructions.