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

ABSTRACT.  Fundamental symmetries of the defocusing nonlinear 
Schr\"odinger 
equation are expressed in action-angle coordinates and 
characterized in terms 
of periodic and Dirichlet spectrum of the associated 
Zakharov-Shabat 
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 
$k=0$.