Y. Shen, P. G. Kevrekidis, N. Whitaker, Boris A. Malomed
We introduce a general model which augments the one-dimensional nonlinear
Schr\"{o}dinger (NLS) equation by nonlinear-diffraction terms competing with
the linear diffraction. The new terms contain two irreducible parameters and
admit a Hamiltonian representation in a form natural for optical media. The
equation serves as a model for spatial solitons near the supercollimation point
in nonlinear photonic crystals. In the framework of this model, a detailed
analysis of the fundamental solitary waves is reported, including the
variational approximation (VA), exact analytical results, and systematic
numerical computations. The Vakhitov-Kolokolov (VK) criterion is used to
precisely predict the stability border for the solitons, which is found in an
exact analytical form, along with the largest total power (norm) that the waves
may possess. Past a critical point, collapse effects are observed, caused by
suitable perturbations. Interactions between two identical parallel solitary
beams are explored by dint of direct numerical simulations. It is found that
in-phase solitons merge into robust or collapsing pulsons, depending on the
strength of the nonlinear diffraction.
View original:
http://arxiv.org/abs/1202.1273
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