Download On a solution of the Sturm-Liouville and the Korteweg-de

Title: On a solution of the Sturm-Liouville and the Korteweg-de-Vries
equations with periodic and almost periodic parameters
using theory of vessels
Abstract:
Using theory of vessels there will be presented a solution of the
Sturm-Liouville equation with a spectral parameter \lambda:
- y''(x) + q(x) y(x) = \lambda^2 y(x).
The construction and existence of a vessel, solving this equation is
closely connected to self-adjoint operators, related to this problem
with different boundary conditions. We will state the main theorem of
vessel existence in the case of periodic and almost periodic q(x).
Using the notion of evolutionary vessel, we will construct a solution
for the KdV equation
q'_t = -3/2 qq'_x +1/4 q'''_{xxx}
with a given periodic/ almost periodic initial value q(x,0).
Bibliography: 1 (previous work)
L. D. Fadeev. The inverse problem in the quantum theory of
scattering. Journal of Mathematical Physics, 4(1):72--104, 1963
B. M. Levitan, I. M. Gelfand. On the determination of a
differential equation from its spectral function (Russian). Izvestiya
Akad. Nauk SSSR. Ser. Mat., 15, 1951.
Levitan B. M. Sargsjan I.S. Introduction to spectral theory:
selfadjoint ordinary differential operators, volume~39. Translations
of mathematical Monographs, Providence, R.I., 1975.
2. (related to new results):
A. Melnikov. Finite dimensional Sturm Liouville vessels and their
tau functions. IEOT 74(4): 455--490, 2011.
A. Melnikov. On a theory of vessels and the inverse scattering.
http://arxiv.org/abs/1103.2392
A. Melnikov. Solution of the KdV equation using evolutionary
vessels. http://arxiv.org/abs/1110.3495
A. Melnikov. On a solution of the Sturm-Liouville and the
Korteweg-de-Vries equations with periodic and almost periodic
parameters using theory of vessels, in preparation