Mobility of solitons in one-dimensional lattices with the cubic-quintic nonlinearity

Cristian Mejía-Cortés, R.A. Vicencio and Boris Malomed


We investigate mobility regimes for localized modes in the discrete nonlinear Schr¨odinger (DNLS) equation
with the cubic-quintic on-site terms. Using the variational approximation, the largest soliton’s total power
admitting progressive motion of kicked discrete solitons is predicted by comparing the effective kinetic energy
with the respective Peierls-Nabarro (PN) potential barrier. The prediction, for theDNLSmodelwith the cubic-only
nonlinearity too, demonstrates a reasonable agreement with numerical findings. A small self-focusing quintic
term quickly suppresses the mobility. In the case of the competition between the cubic self-focusing and quintic
self-defocusing terms, we identify parameter regions where odd and even fundamental modes exchange their
stability, involving intermediate asymmetric modes. In this case, stable solitons can be set in motion by kicking,
so as to let them pass the PN barrier. Unstable solitons spontaneously start oscillatory or progressive motion, if
they are located, respectively, below or above a mobility threshold. Collisions between moving discrete solitons,
at the competing nonlinearities frame, are studied too.