Exact first-order solutions of the nonlinear Schrödinger equation

A method is proposed for finding exact solutions of the nonlinear Schroedinger equations. 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 Schroedinger equation. The obtained solutions form a three-parameter 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.