Breather Mobility and the Peierls-Nabarro Potential: Brief Review and Recent Progress

Abstract

The question whether a nonlinear localized mode (discrete soliton/breather) can be mobile in a lattice has a standard interpretation in terms of the PeierlsNabarro (PN) potential barrier. For the most commonly studied cases, the PN barrier for strongly localized solutions becomes large, rendering these essentially immobile. Several ways to improve the mobility by reducing the PN-barrier have been proposed during the last decade, and the first part gives a brief review of such scenarios in 1D and 2D. We then proceed to discuss two recently discovered novel mobility scenarios. The first example is the 2D Kagome lattice, where the existence of a highly degenerate, flat linear band allows for a very small PN-barrier and mobility of highly localized modes in a small-power regime. The second example is a 1D waveguide array in an active medium with intrinsic (saturable) gain and damping, where exponentially localized, travelling discrete dissipative solitons may exist as stable attractors. Finally, using the framework of an extended Bose-Hubbard model, we show that while quantum fluctuations destroy the mobility of slowly moving, strongly localized classical modes, coherent mobility of rapidly moving states survives even in a strongly quantum regime.

Nonlinear localized modes in dipolar Bose–Einstein condensates in two-dimensional optical lattices

“…A recent review on mobility in discrete nonlinear lattices can be found in Ref. [29]. Based on recent results on dipolar one-dimensional lattices [30], showing similar saturation features, a good mobility may be expected for 2D dipolar lattices as well.…”

“…5, where we have included the respective results for the 1s, 2s and 4s modes for the sake of comparison. The maximum eigenvalue G of the stability matrix is now plotted as a function of the Norm N, in order to make clearer its effect on the dynamics of localized solutions (studied below), which involves stationary solutions at a fixed Norm [29]. From the point where the 1s solution becomes unstable (N ≈ 11), a second IS is found (dotted line in Fig.…”

Asymptotic analysis of breather modes in a two-dimensional mechanical lattice

“…Determining the frequency from (3.7), we obtain ω 2 { opt ac } (0, 0) = Ω 2 + K 2 γ + 1 2 ± γ 2 + 1 4 cos 2 (2ψ) , (A. 19) so the frequencies are well-defined in this limit as ω = Ω; however, the leading order expressions for the ratio of vertical to horizontal displacements, C, as given by (3.8) Since both these expressions depend on ψ at leading order in K ≪ 1, the value of C is not well-defined for either acoustic or optical modes. Furthermore, whilst Next, we consider the behaviour of the system in the limit of K → 0 + (0 < K ≪ 1) where k = π + K cos ψ, and l = π + K sin ψ, (A.…”

“…This concept has been generalised by Vicencio and Johannson [30] to two-dimensional lattices. In [19], Johannson and Jason extend these ideas to breathers in discrete 2D NLS, finding a PN potential which depends on the direction of motion through the lattice. The idea of a PN potential has also been generalised to cover breathers in one-dimensional systems [32], where it is found to depend on the internal phase of the breather.…”

“…Plots of the velocities (3.24)-(3.25) with Ω = 1, γ = 1; left panels show the results for the optical mode; and the right panels illustrate the acoustic mode; horizontal velocities U on the top row, vertical velocities, V , on the bottom row. The Fredholm alternative A 2 + B 2 C = 0 which results in (3.21)-(3.25) only provides a partial solution to (3.17)-(3 19),. in that it means that the two equations are multiples of each other.…”