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

### Abstract

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 . Then there exists a closed subset and a KdV vessel, defined on Ω. For each one can find T x > 0 such that {x} × [ − T x , T x ]⊆Ω. 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 , 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.

© 2014 AIP Publishing LLC

Received 12 August 2013
Accepted 02 October 2014
Published online 22 October 2014

Article outline:

I. INTRODUCTION
II. BACKGROUND ON KREIN SPACE THEORY
III. NON-SYMMETRIC VESSELS
A. Node, prevessel, and vessel
B. Standard construction of a prevessel
C. The tau function of a prevessel
D. Moments and their properties
IV. STURM-LIOUVILLE VESSELS
A. Realizing a function with given moments
B. Construction of a vessel, realizing a given analytic potential
V. KDV EVOLUTIONARY VESSELS
VI. REMARKS

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2014-10-22

2016-09-27

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