 
Summary: MODULATION INSTABILITY AND PERIODIC SOLUTIONS OF THE
NONLINEAR SCHRODINGER EQUATION
N. N. Akhmediev and V. I. Korneev
A very simple exact analytic solution of the nonlinear Schr6dinger
equation is found in the class of periodic solutions. It describes
the time evolution of a wave with constant amplitude on which a small
periodic perturbation is superimposed. Expressions are obtained for
the evolution of the spectrum of this solution, and these expressions
are analyzed qualitatively. It is shown that there exists a certain
class of periodic solutions for which the real and imaginary parts
are linearly related, and an example of a oneparameter family of
such solutions is given.
In recent years, much attention in the scientific literature has been devoted to
periodic solutions of nonlinear partial differential equations [I] and, in particular, the
nonlinear Schr6dinger equation (NSE) (see [25] and the bibliography given there). This is
due to the fact that periodic solutions are needed in a number of practically important
problems, for example, the problem of the generation of picosecond pulses in an optical
fiber [6,7], in the problem of selffocusing [8,9], in the theory of waves on deep water
[i0,ii], and in many other cases. In the class of periodic solutions of the NSE a particular
position is occupied by the solution describing modulation instability, i.e., growth of
