### 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) |
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Pages (from-to) | 148-156 |

Number of pages | 9 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 333 |

DOIs | |

State | Published - Oct 15 2016 |

### Keywords

- Integrability
- Schrödinger operator
- Solitonic gas

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

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## Cite this

*Physica D: Nonlinear Phenomena*,

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