The question whether a nonlinear localized mode (discrete soliton/breather) can be mobile in a lattice has a standard interpretation in terms of the Peierls-Nabarro (PN) potential barrier. In particular, for any system modelled by a Discrete Nonlinear Schrödinger (DNLS) type equation, this concept can be defined as the maximum difference in energy (Hamiltonian) between solutions at fixed power (norm), centered at different lattice positions. For the most commonly studied case with on-site, cubic (Kerr) nonlinearity, the PN barrier for strongly localized solutions becomes large, rendering these essentially immobile.
Several ways to improve the mobility by reducing the PN-barrier for strongly localized modes have been proposed during the last decade, and the first part of this talk will give a brief review of two such scenarios. In 1D, one option is to utilize a competition between on-site and inter-site nonlinearities. In 2D, the mobility is normally much worse than in 1D, due to the fact that also broad solitons are prone to excitation thresholds and quasicollapse instabilities. Utilizing a saturable nonlinearity was found to considerably improve the 2D mobility by reducing the PN barrier in certain parameter regimes for large power.
We then proceed to discuss two (if time allows) recently discovered novel mobility scenarios. The first example discussed is the 2D Kagome lattice, where the existence of a highly degenerate, flat linear band allows small-power, strongly localized nonlinear modes to appear without excitation threshold. The nonlinearity lifts the degeneracy of linear modes and causes a small energy shift between modes centered at different lattice positions, yielding a very small PN-barrier and mobility of highly localized modes in a small-power regime.
The second example discusses a 1D waveguide array in an active medium with intrinsic (saturable) gain and damping. It is shown that exponentially localized, travelling discrete dissipative solitons may exist as stable attractors, supported only by intrinsic properties of the medium (i.e., in absence of any external field or symmetry-breaking perturbations). With a standard, on-site Kerr-nonlinearity the solitons are pinned by the PN-barrier, but decreasing the barrier with inter-site nonlinearities allows for the existence of breathing (i.e., with oscillating size) solitons as stable attractors at certain velocities, related to lattice commensurability effects. The stable moving breathers also survive in presence of weak disorder.
2013. 24-25 p.
Peierls-Nabarro potential, discrete flat-band solitons, discrete dissipative solitons
Quodons in Mica 2013, Nonlinear localized travelling excitations in crystals, Altea, Alicante, Spain, September 18-21, 2013, Meeting in honour of Prof. Francis Michael Russell