Exact first-order solutions of the nonlinear Schrödinger equation

A method of obtaining exact solutions of the nonlinear Schrödinger equation (NSE) is suggested which is based on the substitution connecting real and imaginary parts of the solution by a linear relationship with coefficients depending on time only. The method is essentially the construction of a certain system of ordinary differential equations the solutions of which determine the solutions of NSE. The solutions obtained form a three-parameter family and are expressed in terms of the Jacobi elliptic functions and the third kind incomplete elliptic integral. In general case, the solutions are periodic in spatial variable and double-periodic in time. Particular cases for which the solutions can be expressed in terms of the Jacobi functions and elementary functions are studied in detail. Possibilities of practical applications of the solutions found are pointed out.