Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum ⋯<λ k - ≤λ k + <λ k+1 - ≤⋯ of a Zakharov-Shabat operator is symmetric, i.e. λ k ± =-λ -k ∓ for all k, if and only if the sequence (γ k ) k∈ℤ of gap lengths, γ k :=λ k + -λ k - , is symmetric with respect to k=0.

Grébert, B; Kappeler, T (2002). *Symmetries of the nonlinear Schrödinger equation.* Societe Mathematique de France. Bulletin, 130(4):603-618.

## Abstract

Symmetries of the defocusing nonlinear Schrödinger equation are expressed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Zakharov-Shabat system. Application: proof of the conjecture that the periodic spectrum ⋯<λ k - ≤λ k + <λ k+1 - ≤⋯ of a Zakharov-Shabat operator is symmetric, i.e. λ k ± =-λ -k ∓ for all k, if and only if the sequence (γ k ) k∈ℤ of gap lengths, γ k :=λ k + -λ k - , is symmetric with respect to k=0.

## Citations

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | NLS equation, Zakharov-Shabat operators, action-angle variables, symmetries |

Language: | English |

Date: | 2002 |

Deposited On: | 18 Feb 2010 13:13 |

Last Modified: | 05 Apr 2016 13:25 |

Publisher: | Societe Mathematique de France |

ISSN: | 0037-9484 |

Additional Information: | © 2002 SMF |

Official URL: | http://smf4.emath.fr/Publications/Bulletin/130/html/smf_bull_130_603-618 |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1947455 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1044.35088 |

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