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Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporalstructures
observed in a broad range of applications and natural phenomena. Indeed, they have been predicted
or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion
regime with strong nonlinearity, optical amplifiers, and mode-locked lasers. These self-similar structures are
typically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physical
effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation
with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic
profiles: the zero-dispersion similariton. The zero-dispersion similariton is the final solution state of the system,
not a long-time, intermediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed
by a nonlinear diffusion equation with a rapidly-varying, mean-zero diffusion coefficient. Indeed, the leadingorder
behavior is shown to be governed by the porous media (nonlinear diffusion) equation whose solution
is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating
sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the
zero-dispersion similariton which is, to leading-order, of the Barenblatt form. This is the first analytic model
proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau
model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying
physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed
in mode-locked laser cavities.
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