### 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 language | English (US) |
---|---|

Pages (from-to) | 148-156 |

Number of pages | 9 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 333 |

DOIs | |

State | Published - Oct 15 2016 |

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### Keywords

- Integrability
- Schrödinger operator
- Solitonic gas

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Condensed Matter Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*333*, 148-156. https://doi.org/10.1016/j.physd.2016.04.002

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

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 333, pp. 148-156. https://doi.org/10.1016/j.physd.2016.04.002

}

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 -