## Abstract

Using the Kantorovitch method in combination with a Gaussian ansatz, we derive the equations of motion for spatial, temporal, and spatiotemporal optical propagation in a dispersive Kerr medium with a general transverse and spectral gain profile. By rewriting the variational equations as differential equations for the temporal and spatial Gaussian *q* parameters, optical $ABCD$ matrices for self-focusing and self-phase modulation, a general transverse gain profile, and nonparabolic spectral gain filtering are obtained. Further effects can easily be taken into account by adding the corresponding $ABCD$ matrices. Applications include the temporal pulse dynamics in gain fibers and the beam propagation or spatiotemporal pulse evolution in bulk gain media. As an example, the steady-state spatiotemporal Gaussian pulse dynamics in a Kerr-lens mode-locked laser resonator is studied.

© 2006 Optical Society of America

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