Maximum amplitudes of finite-gap solutions for the focusing Nonlinear Schr\"odinger Equation

In this paper we prove that the maximum amplitude of a finite-gap solution to the focusing Nonlinear Schr"{o}dinger equation with given spectral bands does not exceed half of the sum of the length of all the bands. This maximum will be attained for certain choices of the initial phases. A similar result is also true for the defocusing Nonlinear Schr"{o}dinger equation.