Symmetries of the nonlinear Schrödinger equation

Benoît Grébert; Thomas Kappeler

Bulletin de la Société Mathématique de France (2002)

  • Volume: 130, Issue: 4, page 603-618
  • ISSN: 0037-9484

Abstract

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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 .

How to cite

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Grébert, Benoît, and Kappeler, Thomas. "Symmetries of the nonlinear Schrödinger equation." Bulletin de la Société Mathématique de France 130.4 (2002): 603-618. <http://eudml.org/doc/272455>.

@article{Grébert2002,
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 $\cdots &lt; \lambda ^-_k \le \lambda ^+_k &lt; \lambda ^-_\{k + 1\} \le \cdots $ of a Zakharov-Shabat operator is symmetric,i.e. $\lambda ^\pm _k = - \lambda ^\mp _\{-k\}$ for all $k$, if and only if the sequence $(\gamma _k)_\{k\in \mathbb \{Z\}\}$ of gap lengths, $\gamma _k:= \lambda ^+_k - \lambda ^-_k$, is symmetric with respect to $k=0$.},
author = {Grébert, Benoît, Kappeler, Thomas},
journal = {Bulletin de la Société Mathématique de France},
keywords = {NLS equation; Zakharov-Shabat operators; action-angle variables; symmetries},
language = {eng},
number = {4},
pages = {603-618},
publisher = {Société mathématique de France},
title = {Symmetries of the nonlinear Schrödinger equation},
url = {http://eudml.org/doc/272455},
volume = {130},
year = {2002},
}

TY - JOUR
AU - Grébert, Benoît
AU - Kappeler, Thomas
TI - Symmetries of the nonlinear Schrödinger equation
JO - Bulletin de la Société Mathématique de France
PY - 2002
PB - Société mathématique de France
VL - 130
IS - 4
SP - 603
EP - 618
AB - 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 $\cdots &lt; \lambda ^-_k \le \lambda ^+_k &lt; \lambda ^-_{k + 1} \le \cdots $ of a Zakharov-Shabat operator is symmetric,i.e. $\lambda ^\pm _k = - \lambda ^\mp _{-k}$ for all $k$, if and only if the sequence $(\gamma _k)_{k\in \mathbb {Z}}$ of gap lengths, $\gamma _k:= \lambda ^+_k - \lambda ^-_k$, is symmetric with respect to $k=0$.
LA - eng
KW - NLS equation; Zakharov-Shabat operators; action-angle variables; symmetries
UR - http://eudml.org/doc/272455
ER -

References

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  1. [1] J. Bourgain – « Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations », GAFA3 (1993), p. 107–156. Zbl0787.35097MR1209299
  2. [2] L. Faddeev & L. Takhtajan – Hamiltonian methods in the theory of solitons, Springer, 1987. Zbl1111.37001MR905674
  3. [3] B. Grébert – « Problèmes spectraux inverses pour les systèmes AKNS sur la droite réelle », Thèse, Université Paris-Nord, 1990. 
  4. [4] B. Grébert & J. Guillot – « Gaps of one dimensional periodic AKNS systems », Forum Math5 (1993), p. 459–504. Zbl0784.34024MR1232720
  5. [5] B. Grébert & T. Kappeler – « Perturbations of the NLS equation », to appear in Milan J. of Math. Zbl1048.37067MR2120919
  6. [6] —, « Théorème de type KAM pour l’équation de Schrödinger non linéaire », 327 (1998), p. 473–478. Zbl0913.35125MR1652566
  7. [7] B. Grébert, T. Kappeler & J. Pöschel – « Normal form theory for NLS », preliminary version available (please contact authors). Zbl1298.35002
  8. [8] H. McKean & K. Vaninsky – « Action-angle variables for the cubic Schrödinger equation », CPAM50 (1997), p. 489–562. Zbl0990.35047MR1441912

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