 99221 B. Grebert, T. Kappeler
 Symmetries of the Nonlinear Schr\"odinger Equation
(53K, Latex)
Jun 7, 99

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Abstract. Fundamental symmetries of the defocusing nonlinear
Schr\"odinger
equation are expressed in actionangle coordinates and
characterized in terms
of periodic and Dirichlet spectrum of the associated
ZakharovShabat
system. As a main application we prove a conjecture, raised by
several experts in field, that the periodic spectrum is symmetric
iff the sequence of gap lengths $(\gamma_k)_{k\in \mathbb {Z}}$ or, equivalently, the
sequence of actions $(I_k)_{k\in \mathbb {Z}}$ is symmetric with respect to
$k=0$.
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