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Yu.S. Kivshar and D.E. Pelinovsky

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Self-focusing and transverse instabilities of solitary waves

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Physics Reports 331, 118-195 (2000)

We give an overview of the basic physical concepts
and analytical methods for investigating the
symmetry-breaking instabilities of solitary waves.
We discuss self-focusing of
spatial optical solitons in diffractive nonlinear
media due to either transverse (one more unbounded
spatial dimension) or modulational (induced by
temporal wave dispersion)
instabilities, in the framework of the cubic nonlinear
Schrodinger (NLS) equation and its generalizations.
Both linear and nonlinear regimes of the instability-induced soliton
dynamics are analyzed for bright (self-focusing media)
and dark (self-defocusing media) solitary waves.
For a defocusing Kerr medium, the results of the
small-amplitude limit
are compared with the theory of the transverse
instabilities of the Korteweg-de Vries solitons
developed in the framework of the exactly integrable
Kadomtsev-Petviashvili
equation. We give also a comprehensive summary of
different physical problems involving the analysis
of the transverse and modulational instabilities
of solitary waves including
the soliton self-focusing in the discrete NLS
equation, the models of parametric wave mixing,
the Davey-Stewartson equation, the Zakharov-Kuznetsov
and Shrira equations,
instabilities of higher-order and ring-like spatially
localized modes, the kink stability in the dissipative
Cahn-Hilliard equation, etc. Experimental observations
of the soliton
self-focusing and transverse instabilities for bright
and dark solitons in nonlinear optics are briefly
summarized as well.

**Keywords:**

NONLINEAR SCHRODINGER-EQUATION, CAHN-HILLIARD EQUATION,
KADOMTSEV-PETVIASHVILI EQUATION, ZAKHAROV-KUZNETSOV EQUATION,
NORMALLY DISPERSIVE MEDIA, ION-ACOUSTIC-WAVES,
PHOTOREFRACTIVE SCREENING SOLITONS, FEMTOSECOND PULSE-PROPAGATION,
DIMENSIONAL KINK SOLUTION, OPTICAL VORTEX SOLITONS