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D. Pelinovsky

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Translationally invariant nonlinear Schrodinger lattices

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Nonlinearity 19, 2695-2716 (2006)

**Abstract:**

Persistence of stationary and traveling single-humped localized
solutions in the spatial discretizations of the nonlinear
Schrodinger (NLS) equation is addressed. The discrete NLS
equation with the most general cubic polynomial function is
considered. Constraints on the nonlinear function are found from the
condition that the second-order difference equation for stationary
solutions can be reduced to the first-order difference map. The
discrete NLS equation with such an exceptional nonlinear function is
shown to have a conserved momentum but admits no standard
Hamiltonian structure. It is proved that the reduction to the
first-order difference map gives a sufficient condition for
existence of translationally invariant single-humped stationary
solutions and a necessary condition for existence of single-humped
traveling solutions. Other constraints on the nonlinear function are
found from the condition that the differential advance-delay
equation for traveling solutions admits a reduction to an integrable
normal form given by a third-order differential equation. This
reduction also gives a necessary condition for existence of
single-humped traveling solutions. The nonlinear function which
admits both reductions defines a two-parameter family of discrete
NLS equations which generalizes the integrable Ablowitz-Ladik
lattice.

**Keywords**:

DISCRETE NLS EQUATIONS, TRAVELLING WAVES, TRANSLATIONALLY INVARIANT
LATTICES, NORMAL FORMS, CENTER MANIFOLD REDUCTIONS,
THIRD-ORDER DERIVATIVE NLS EQUATION, EMBEDDED SOLITONS,
BIFURCATIONS OF NONLINEAR WAVES