Continuous description of lattice discreteness effects in front propagation

Abstract

Models describing microscopic or mesoscopic phenomena in physics are inherently discrete, where the lattice spacing between fundamental components, such as in the case of atomic sites, is a fundamental physical parameter. The effect of spatial discreteness over front propagation phenomenon in an overdamped one-dimensional periodic lattice is studied. We show here that the study of front propagation leads in a discrete description to different conclusions that in the case of its, respectively, continuous description, and also that the results of the discrete model, can be inferred by effective continuous equations with a supplementary spatially periodic term that we have denominated Peierls-Nabarro drift, which describes the bifurcation diagram of the front speed, the appearance of particle-type solutions and their snaking bifurcation diagram. Numerical simulations of the discrete equation show quite good agreement with the phenomenological description.

Transfer of dipolar gas through the discrete localized mode

“…The DB or ILM emerges due to the interplay between the nonlinearity and discreteness of the system. The DB or ILM has been observed in various systems, such as Josephson-junction arrays [8,9], antiferromagnet systems [10,11], micromechanical cantilever arrays [12], nonlinear waveguide arrays [13,14], Bose-Einstein condensate in optical lattices [15,16], Tonks gas in optical lattice [17], superfluid Fermi gases in optical lattices [18], and some dissipative systems [19,20], etc. The static, dynamical, and other properties of DBs or ILMs for nondipolar condensates have been studied theoretically throughout the last decades [21][22][23][24][25][26].…”

Plasmonic kinks and walking solitons in nonlinear lattices of metal nanoparticles

“…This underlines a common nature of these nonlinear modes since discrete solitons can be interpreted as two tightly bound kinks with opposite polarities. Additionally, this soliton family can be treated as a consequence of the so-called snaking bifurcation considered earlier in Ginzburg-Landau models [31,33].…”

“…Constant E ⊥ 0 results in a constant kink velocity, v. That is why a step-like time dependence of E ⊥ 0 allows us to control v. The velocity sign is defined as positive if the motion of the switching wave leads to the expansion of the region occupied by the upper branch of bistability; otherwise, the velocity is marked as negative. Hereinafter, we treat kink and soliton velocities as time-averaged values because the discrete nature of nanoparticle arrays leads to small-amplitude harmonic oscillations in the instantaneous value of v [31].…”

Nonlinear dynamics in meso and nano scales: fundamental aspects and applications

“…Wio et al [80] use the idea of non-equilibrium potentials and a stochastic nonlinear partial differential equation, known as Kardar-Parisi-Zhang, to phenomenologically discuss surface and interface growth in mesoscopic systems. The effect of spatial discreteness-for instance, present in atomic lattices-on the wave-front propagation that is under the action of damping is described by Clerc et al [81] by means of continuous differential equations. Masoller & Rosso [82] consider a way to quantify statistical complexity (useful in characterizing pattern formation in quantum nano and meso problems) for a prototype system, the (delayed) logistic map with nonlinear feedback.…”