Symmetries of the nonlinear Schrodinger equation

Symmetries of the defocusing nonlinear Schrodinger 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... <lambda(k)(-) less than or equal to lambda(k)(+) < lambda(k+1)(-) less than or equal to ... of a Zakharov-Shabat operator is symmetric, i.e. lambda(k)(+/-) = -lambda(-k)(-/+) for all k, if and only if the sequence (gamma(k))(kis an element ofZ) of gap lengths, gamma(k) := lambda(k)(+) - lambda(k)(-), is symmetric with respect to k = 0.