2012/05/23 by Andrey Melnikov, Melnikov, Andrey
Mathematics · Physics and Astronomy · #30B60 #34B05 #34B15 #34B24 #35G31 #35N30 #35P25 #35P30 #46N20 #47E05 #47F05 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.AP #math.CA #math.FA #math.MP #math.SP #msc:30B60 #msc:34B05 #msc:34B15 #msc:34B24 #msc:35G31 #msc:35N30 #msc:35P25 #msc:35P30 #msc:46N20 #msc:47E05 #msc:47F05
paper · pdf · doi:10.48550/arxiv.1205.5285
This paper has been withdrawn by the author due to a crucial error in formulas
arxiv created 2012/12/08 · arxiv updated 2012/12/11
We prove the existence of solutions to the Sturm-Liouville (SL) equation -y"(x)+q(x)y(x) = s2 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.