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Self-Focusing and Collapse of Light
Beams in Nonlinear Dispersive Media
Luc Berg´e
Jens Juul Rasmussen
ABSTRACT The collapse of self-focusing beams in nonlinear dispersive
media described by the nonlinear Schr¨odinger (NLS)equation is reviewed.
Conditions for blow-up of solutions to the NLS equation with a cubic non-
linearity and isotropic dispersion properties are recalled, together with the
self-similar analyses employed for modelling wave collapses. Emphasis is
then laid on the influence of anisotropic (negative)dispersion and on the
deviations from the spatio-temporal envelope approximations, which are
shown to strongly alter the blow-up dynamics.
1 Introduction
The self-focusing and collapse of wave-packets in nonlinear dispersive media
as, ., plasmas and nonlinear optical materials, is in general described
by the nonlinear Schr¨odinger (NLS) equation for a scalar wave envelope
E(x, y, z, t).In normalized form this equation reads
2
∂E ∂ E 2
i +∆⊥E + s + |E| E =0. ()
∂t ∂z2
The different signs of the coefficient s allow for treating media with isotropic
dispersion (s>0) as well as with anisotropic dispersion (s<0).In
Eq.() we have used standard notations and normalized variables. In
the context of nonlinear optics, the variable t often refers to the longitu-
dinal length along the beam propagation axis, whereas z occurring in the
space derivatives of the envelope corresponds to a retarded time variable
t = t − z/ω (ω ≡∂ω/∂k).These derivatives then reflect the “temporal”
wave dispersion measured through the group velocity dispersion (GVD)
2 2|
coefficient s, related to the dispersion factor ∂ k/∂ωω0 .The coefficient
s can be either positive in the case of a so-called anomalous dispersion,
or negative in the opposite case of a so-called normal dispersion [1].More
generally, equation () applies to the description of nonlinear wave prop-
agation in media with an ani