### 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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Title of host publication | Trends in Mathematics |

Publisher | Springer International Publishing |

Pages | 221-233 |

Number of pages | 13 |

Edition | 9783319635934 |

DOIs | |

State | Published - Jan 1 2018 |

### Publication series

Name | Trends in Mathematics |
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Number | 9783319635934 |

ISSN (Print) | 2297-0215 |

ISSN (Electronic) | 2297-024X |

### Keywords

- Integrable turbulence
- KdV equation
- Riemann–Hilbert problem
- Schrödinger operator

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Trends in Mathematics*(9783319635934 ed., pp. 221-233). (Trends in Mathematics; No. 9783319635934). Springer International Publishing. https://doi.org/10.1007/978-3-319-63594-1_22