Letters in Mathematical Physics

, Volume 106, Issue 6, pp 731–740 | Cite as

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

  • Dmitry V. Zakharov
  • Sergey A. Dyachenko
  • Vladimir E. Zakharov


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.


integrable systems Schrödinger operator soliton solutions 

Mathematics Subject Classification

70H06 81U15 


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Dmitry V. Zakharov
    • 1
  • Sergey A. Dyachenko
    • 2
  • Vladimir E. Zakharov
    • 3
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsUniversity of Illinois at Urbana–ChampaignUrbanaUSA
  3. 3.Department of MathematicsUniversity of ArizonaTucsonUSA

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