TY - JOUR

T1 - Bounded Solutions of KdV and Non-Periodic One-Gap Potentials in Quantum Mechanics

AU - Zakharov, Dmitry V.

AU - Dyachenko, Sergey A.

AU - Zakharov, Vladimir E.

N1 - Publisher Copyright:
© 2016, Springer Science+Business Media Dordrecht.

PY - 2016/6/1

Y1 - 2016/6/1

N2 - We describe a broad new class of exact solutions of the KdV hierarchy. In general, these solutions do not vanish at infinity, and are neither periodic nor quasi-periodic. This class includes algebro-geometric finite-gap solutions as a particular case. The spectra of the corresponding Schrödinger operators have the same structure as those of N-gap periodic potentials, except that the reflectionless property holds only in the infinite band. These potentials are given, in a non-unique way, by 2N real positive functions defined on the allowed bands. In this letter we restrict ourselves to potentials with one allowed band on the negative semi-axis; however, our results apply in general. We support our results with numerical calculations.

AB - We describe a broad new class of exact solutions of the KdV hierarchy. In general, these solutions do not vanish at infinity, and are neither periodic nor quasi-periodic. This class includes algebro-geometric finite-gap solutions as a particular case. The spectra of the corresponding Schrödinger operators have the same structure as those of N-gap periodic potentials, except that the reflectionless property holds only in the infinite band. These potentials are given, in a non-unique way, by 2N real positive functions defined on the allowed bands. In this letter we restrict ourselves to potentials with one allowed band on the negative semi-axis; however, our results apply in general. We support our results with numerical calculations.

KW - Schrödinger operator

KW - integrable systems

KW - soliton solutions

UR - http://www.scopus.com/inward/record.url?scp=84963769152&partnerID=8YFLogxK

U2 - 10.1007/s11005-016-0838-6

DO - 10.1007/s11005-016-0838-6

M3 - Article

AN - SCOPUS:84963769152

VL - 106

SP - 731

EP - 740

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 6

ER -