Primitive potentials and bounded solutions of the KdV equation

S. Dyachenko, D. Zakharov, Vladimir E Zakharov

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

We construct a broad class of bounded potentials of the one-dimensional Schrödinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.

Original languageEnglish (US)
Pages (from-to)148-156
Number of pages9
JournalPhysica D: Nonlinear Phenomena
Volume333
DOIs
StatePublished - Oct 15 2016

Fingerprint

singular integral equations
hierarchies
turbulence
operators
simulation

Keywords

  • Integrability
  • Schrödinger operator
  • Solitonic gas

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Condensed Matter Physics

Cite this

Primitive potentials and bounded solutions of the KdV equation. / Dyachenko, S.; Zakharov, D.; Zakharov, Vladimir E.

In: Physica D: Nonlinear Phenomena, Vol. 333, 15.10.2016, p. 148-156.

Research output: Contribution to journalArticle

@article{584d1d1998e74ccda8bce2ecd2c0f125,
title = "Primitive potentials and bounded solutions of the KdV equation",
abstract = "We construct a broad class of bounded potentials of the one-dimensional Schr{\"o}dinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive H{\"o}lder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.",
keywords = "Integrability, Schr{\"o}dinger operator, Solitonic gas",
author = "S. Dyachenko and D. Zakharov and Zakharov, {Vladimir E}",
year = "2016",
month = "10",
day = "15",
doi = "10.1016/j.physd.2016.04.002",
language = "English (US)",
volume = "333",
pages = "148--156",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",

}

TY - JOUR

T1 - Primitive potentials and bounded solutions of the KdV equation

AU - Dyachenko, S.

AU - Zakharov, D.

AU - Zakharov, Vladimir E

PY - 2016/10/15

Y1 - 2016/10/15

N2 - We construct a broad class of bounded potentials of the one-dimensional Schrödinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.

AB - We construct a broad class of bounded potentials of the one-dimensional Schrödinger operator that have the same spectral structure as periodic finite-gap potentials, but that are neither periodic nor quasi-periodic. Such potentials, which we call primitive, are non-uniquely parametrized by a pair of positive Hölder continuous functions defined on the allowed bands. Primitive potentials are constructed as solutions of a system of singular integral equations, which can be efficiently solved numerically. Simulations show that these potentials can have a disordered structure. Primitive potentials generate a broad class of bounded non-vanishing solutions of the KdV hierarchy, and we interpret them as an example of integrable turbulence in the framework of the KdV equation.

KW - Integrability

KW - Schrödinger operator

KW - Solitonic gas

UR - http://www.scopus.com/inward/record.url?scp=84992312368&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84992312368&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2016.04.002

DO - 10.1016/j.physd.2016.04.002

M3 - Article

AN - SCOPUS:84992312368

VL - 333

SP - 148

EP - 156

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -