 
Summary: EXACT FIRSTORDER SOLUTIONS OF THE NONLINEAR SCHRODINGER EQUATION
N. N. Akhmediev, V. M. Eleonskii, and
N. E. Kulagin
A method is proposed for finding exact solutions of the nonlinear
Schr~dinger equation. It uses an ansatz in which the real and
imaginary parts of the unknown function are connected by a linear
relation with coefficients that depend only on the time. The
method consists of constructing a system of ordinary differential
equations whose solutions determine solutions of the nonlinear
Schr~dinger equation. The obtained solutions form a threeparameter
family that can be expressed in terms of elliptic Jacobi functions and
the incomplete elliptic integral of the third kind. In the general
case, the obtained solutions are periodic with respect to the spatial
variable and doubly periodic with respect to the time. Special cases
for which the solutions can be expressed in terms of elliptic Jacobi
functions and elementary functions are considered in detail. Possible
fields of practical applications of the solutions are mentioned.
I. Introduction
The nonlinear SchrSdinger equation (NSE) is one of the representatives of the class
of completely integrable partial differential equations that has great applied importance.
