### Abstract

We describe a broad new class of exact solutions of the KdV hierarchy. In general, these solutions do not vanish at infinity, and are neither periodic nor quasi-periodic. This class includes algebro-geometric finite-gap solutions as a particular case. The spectra of the corresponding Schrödinger operators have the same structure as those of N-gap periodic potentials, except that the reflectionless property holds only in the infinite band. These potentials are given, in a non-unique way, by 2N real positive functions defined on the allowed bands. In this letter we restrict ourselves to potentials with one allowed band on the negative semi-axis; however, our results apply in general. We support our results with numerical calculations.

Original language | English (US) |
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Pages (from-to) | 731-740 |

Number of pages | 10 |

Journal | Letters in Mathematical Physics |

Volume | 106 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2016 |

### Keywords

- Schrödinger operator
- integrable systems
- soliton solutions

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Letters in Mathematical Physics*,

*106*(6), 731-740. https://doi.org/10.1007/s11005-016-0838-6