T. Dohnal and Pelinovsky
Surface gap solitons at a nonlinearity interface
SIAM Journal of Applied Dynamical Systems, 7, 249-264 (2008)
Abstract:
We demonstrate existence of waves localized at the interface of
two nonlinear periodic media with different coefficients of the
cubic nonlinearity via the one-dimensional Gross--Pitaevsky
equation. We call these waves the surface gap solitons (SGS). In
the case of smooth symmetric periodic potentials, we study
analytically bifurcations of SGS's from standard gap solitons and
determine numerically the maximal jump of the nonlinearity
coefficient allowing for the SGS existence. We show that the
maximal jump vanishes near the thresholds of bifurcations of gap
solitons. In the case of continuous potentials with a jump in the
first derivative at the interface, we develop a homotopy method of
continuation of SGS families from the solution obtained via gluing
of parts of the standard gap solitons and study existence of SGS's
in the photonic band gaps. We explain the termination of the SGS
families in the interior points of the band gaps from the
bifurcation of linear bound states in the continuous non-smooth
potentials.
Keywords:
Gross-Pitaevskii equation, periodic potentials, nonlinearity interface,
gap solitons and surface solitons, existence and stability, bifurcations of solutions