  Self-focusing emerges as a nonlinear optical phenomenon resulting from an
intense laser field and plasma interaction. This study investigates the
self-focusing behavior of Gaussian laser beams within magnetized plasma
environments utilizing a novel approach, source-dependent expansion. By
employing source-dependent expansion, we explore the intricate dynamics of
laser beam propagation, considering the influence of plasma density and
external magnetic fields. The interplay between the beam's Gaussian profile and
the self-focusing mechanism through rigorous mathematical analysis and
numerical simulations, particularly in the presence of plasma-induced
nonlinearities are elucidated here. Our findings reveal crucial insights into
the evolution of laser beams under diverse parameters, the ponderomotive force,
relativistic factors, plasma frequency, polarization states, external magnetic
field, wavelength, and laser intensity. This research not only contributes to
advancing our fundamental understanding of laser-plasma interactions but also
holds promise for optimizing laser-driven applications.
