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P. Panayotaros and D. Pelinovsky

#
Periodic oscillations of discrete NLS solitons in
the presence of diffraction management

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Nonlinearity 21, 1265-1279 (2008)

**Abstract:**

We consider the discrete NLS equation with a small-amplitude
time-periodic diffraction coefficient which models diffraction
management in nonlinear lattices. In the space of one dimension
and at the zero-amplitude diffraction management, multi-peak
localized modes (called discrete solitons or discrete breathers)
are stationary solutions of the discrete NLS equation which are
uniquely continued from the anti-continuum limit, where they are
compactly supported on finitely many non-zero nodes. We prove that
the multi-peak localized modes are uniquely continued to the
time-periodic space-localized solutions for small-amplitude
diffraction management if the period of the diffraction
coefficient is not multiple to the period of the stationary
solution. The same result is extended to multi-peaked localized
modes in the space of two and three dimensions (which include
discrete vortices) under additional non-degeneracy assumptions on
the stationary solutions in the anti-continuum limit.

**Keywords:**

discrete nonlinear Schordinger equation, diffraction management,
persistence of stationary and periodic solutions, Lyapunov Theorem
on periodic orbit, Implicit Function Theorem.