D.E. Pelinovsky, A.A. Sukhorukov, and Yu.S. Kivshar
Bifurcations and stability of gap solitons in periodic potentials
Phys. Rev. E 70, 036618 (2004)
Abstract:
We analyze the existence, stability, and internal modes of
gap solitons in nonlinear periodic systems described by the
nonlinear Schrodinger equation with a sinusoidal potential, such
as photonic crystals, waveguide arrays, optically-induced photonic
lattices, and Bose-Einstein condensates loaded onto an optical
lattice. We study bifurcations of gap solitons from the band edges
of the Floquet-Bloch spectrum, and show that gap solitons can
appear near all lower or upper band edges of the spectrum,
for focusing or defocusing nonlinearity, respectively. We show
that, in general, two types of gap solitons can bifurcate
from each band edge, and one of those two is always unstable.
A gap soliton corresponding to a given band edge is
shown to possess a number of internal modes that bifurcate
from all band edges of the same polarity.
We demonstrate that stability of gap solitons is determined by
location of the internal modes with respect to the spectral bands
of the inverted spectrum and, when they overlap, complex
eigenvalues give rise to oscillatory instabilities of gap
solitons.
Keywords:
NONLINEAR SCHRODINGER EQUATION, PERIODIC POTENTIAL, BOSE-EINSTEIN CONDENSATES,
GAP SOLITONS, BIFURCATIONS, STABILITY, INTERNAL MODES, MELNIKOV INTERGRALS,
EVANS FUNCTION.