##
D.E. Pelinovsky and V.M. Rothos

#
Bifurcations of travelling breathers in the discrete NLS equations

###
Physica D 202, 16-36 (2005)

**Abstract:**

We study discrete NLS equations, which include the cubic NLS
lattice with on-site interactions and the integrable
Ablowitz--Ladik lattice. Standing wave solutions (discrete
breathers) are known to exist in the discrete NLS equations
outside of the spectral band. We study travelling wave solutions
(travelling breathers) which have nonlinear resonances with
unbounded spectral bands.

We apply methods of normal forms and center manifold reductions
and show that a continuous NLS equation with the third-order
derivative term is a canonical normal form for the discrete NLS
equation. Bifurcations of travelling breathers are analyzed in the
framework of the third-order derivative NLS equation.

It is shown that there is a continuous two-parameter family of
single-humped travelling breathers in the third-order derivative
NLS equation, when it is derived from the integrable
Ablowitz-Ladik lattice. On the contrary, there are no
single-humped solutions but there exists an infinite discrete set
of one-parameter families of double-humped travelling breathers in
the third-order derivative NLS equation, when it is derived from
the cubic NLS equation with on-site interactions.

**Keywords:**

DISCRETE NLS EQUATIONS, TRAVELLING BREATHERS, CUBIC DNLS LATTICE,
INTEGRABLE ABLOWITZ-LADIK LATTICE, NORMAL FORMS, CENTER MANIFOLD REDUCTIONS,
THIRD-ORDER DERIVATIVE NLS EQUATION, EMBEDDED SOLITONS,
BIFURCATIONS OF NONLINEAR WAVES