Resonant localized modes in electrical lattices with second-neighbor coupling

Abstract

We demonstrate experimentally and corroborate numerically that an electrical lattice with nearest-neighbor and second-neighbor coupling can simultaneously support long-lived coherent structures in the form of both standard intrinsic localized modes (ILMs), as well as resonant ILMs. In the latter case, the wings of the ILM exhibit oscillations due to resonance with a degenerate plane-wave mode. This kind of localized mode has also been termed nanopteron. Here we show experimentally and using realistic simulations of the system that the nanopteron can be stabilized via both direct and subharmonic driving. In the case of excitations at the zone center (i.e., at wavenumber k = 0), we observed stable ILMs, as well as a periodic localization pattern in certain driving regimes. In the zone boundary case (of wavenumber k = π/a, where a is the lattice spacing), the ILMs are always resonant with a plane-wave mode, but can nevertheless be stabilized by direct (staggered) and subharmonic driving.

Embedded solitons in the double sine-Gordon lattice with next-neighbor interactions

“…This research can be further extended into other physical models that are not connected to the Josephson effect. Recent research on the nonlinear electric circuits with the next-neighbor interactions [35] seems to be a promising field for application of the ideas developed in this article.…”

“…The more correct description of the various nonlinear phenomena in lattices requires consideration of not just the nearest-neighbor interactions but also the nextneighbor and/or the further distant neighbor interactions [7,[33][34][35]. In the case of JJ arrays this means that not only the coupling due to the self-inductance of each JJ cell should be accounted for, but also the mutual inductances between the cells [36] should be taken into consideration.…”

Linear impurity modes in an electrical lattice: Theory and experiment

“…Our physical setup is that of [17,18] (see also [19][20][21] for some recent experimental studies in this system). In particular, we consider the bi-inductive electrical lattice arXiv:1909.10981v1 [nlin.PS] 24 Sep 2019 shown in Fig.…”

“…In the present work, we take advantage of the well established framework of electrical transmission lines [17,18] as a prototypical setting where the theory of linear impurity modes can be tested. Upon modeling the experimental setting of recent experiments such as [19][20][21] * mmolina@uchile.cl at the linear level (i.e., in the absence of nonlinearity), we utilize the formulation of Green's functions [22][23][24] in order to identify both the eigenvalues/eigenfrequencies and eigenfunctions of the linear modes of the lattice, focusing naturally on the localized vibrations thereof. We provide analytical expressions for these and a systematic comparison with the corresponding experimental results.…”

Backward- and forward-wave soliton co-existence due to second-neighbor coupling in a left-handed transmission line

“…In this physical context, we briefly highlight two recent references. The first [16] examined the highly nonlinear regime, where the discreteness of the lattice is essential; the second [17] investigated the more weakly nonlinear limit, where a semi-discrete approximation can be used. The former found resonant ILMs (in addition to the more standard ILMs).…”