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The dynamics that result from the combined effects of spatial diffraction, nonlinearity, and a parabolic graded index for wave propagation in optical fibers are presented. An approximate analytical solution of the nonlinear Schrödinger equation in a graded-index fiber is obtained by using a variational approach. Particular emphasis is put on the variation of both the pulse width and the longitudinal phase delay with the distance of propagation along the fiber.

An investigation is made of the formation of a single soliton, or a pair of solitons, from initially nontransform limited pulses in a nonlinear Kerr medium having anomalous dispersion. A qualitative physical explanation is given for the formation of soliton pairs. Approximate solutions for the amplitudes and the velocities of the generated solitons are established and corroborated by numerical solutions of the Zakharov-Shabat eigenvalue problem.