##
T.R.O. Melvin, A.R. Champneys, and D.E. Pelinovsky

#
Discrete traveling solitons in the Salerno model

###
SIAM Journal of Applied Dynamical Systems 8, 689–709 (2009)

**Abstract:**

We investigate travelling solitary waves in the 1D Salerno model,
which interpolates between the cubic discrete nonlinear Schrodinger
(DNLS) equation and the integrable Ablowitz-Ladik (AL) model.
In a travelling frame the model becomes an advance delay equation to which
we analyse existence of homoclinic orbits to the rest state.
The method of beyond all orders asymptotics is used to
compute the so-called Stokes
constant that measures the splitting of the stable and
unstable manifolds. Through computing zeros of the Stokes constant,
we find that there exists
a number of solution families for parameter values between
the DNLS and AL limits of the Salerno model. Using a
pseudo-spectral method, we
numerically continue these solution families and show
that their parameters approach the curves of the zero
level of the Stokes constant as the
soliton amplitude approaches zero. An interesting
topological structure of solutions occurs in parameter space.
As the AL limit is approached
solutions sheets of single-hump solutions undergo folds
and become double-hump solitons. Numerical simulation
suggest the single-humps to be
stable and to interact almost inelastically.

**Keywords:**

discrete nonlinear Schrodinger equation, traveling wave solutions,
differential advance-delay equations, Stokes constants, beyond-all-order
asymptotic expansions, stability of nonlinear waves