## 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 |
---|---|

Pages (from-to) | 148-156 |

Number of pages | 9 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 333 |

DOIs | |

State | Published - Oct 15 2016 |

Externally published | Yes |

## Keywords

- Integrability
- Schrödinger operator
- Solitonic gas