Discrete embedded solitary waves and breathers in one-dimensional nonlinear lattices

In quantum mechanics it is typically expected that for a particle in the presence of an external potential (whose value at infinity is set as the zero level), the eigenstates consist of either extended modes, via a quasi-continuum spectrum with positive energies, or of localized modes, with negative energy, forming a discrete (so-called point) spectrum. This result, which stems from a general analysis of the Schrödinger equation, was challenged by Wigner and von Neumann [1] who showed that it was possible for a localized state to co-exist with the extended modes inside the continuum band. In a seminal work, they constructed explicitly a three-dimensional potential, using a reverse-engineering approach, to produce a setting that was tailored to support a so-called bound state in the continuum or embedded mode (EM). The method had to face some difficulties since, for instance, the EM produced decreased in space as a power law, making it non-renormalizable. Ideally, one would like to have a bona fide normalized, localized mode that is decoupled from the continuum, like a resonance with zero width.

Regarded at first as a mathematical curiosity, the topic of EM has re-emerged recently with an abundance of works on its theory and applications in many diverse areas where wave phenomena (quantum and classical) are dominant. During the 1970s, Stillinger [2, 3] and Herrick [4] improved the theory of Wigner and von Neumann and pointed out that EMs could be found in certain atomic and molecular systems. They also suggested the use of superlattices to construct potentials that could support EMs [5, 6]. More recently, direct observations of electronic bound states above a potential well and localized by Bragg reflections, were carried out using semiconductor heterostructures [7]. A different approach comes from the physics of resonant states in quantum mechanics. These states are spatially localized but with non-decaying tails and energies inside the band, and they eventually decay, i.e., they possess a finite lifetime. However, it is possible to arrange conditions in order to make a given resonance interfere negatively with another one and produce a zero width resonance, that is, an EM. This has been shown to occur in a hydrogen atom immersed in a magnetic field, modeled as a system of coupled Coulombic channels, where interference between resonances belonging to different channels can lead to the creation of EM [8, 9]. In recent times, EMs have been observed in mesoscopic electron transport and quantum waveguides [10, 11, 12, 13, 14, 15, 16, 17, 18], and in quantum dot systems [19, 20, 21, 22, 23]. A common theme in all the above systems is that the onset of an EM can be traced back to the destruction of the discrete-continuum decay channels by quantum interference effects. The ultimate origin of the EM phenomenon is regarded nowadays as the result of interference and thus it should be inherent to any wavelike theory besides quantum mechanics. The original approach of Wigner and von Neumann has been extended to the case of a discrete and periodic system, such as the usual tight-binding model. There, the methodology consists of choosing an envelope that modulates a given chosen extended eigenstate, and then imposing that its energy coincides with the original one (no envelope). To achieve this a site energy distribution is chosen judiciously, so that the energy for the modulated state coincides with the energy of the original mode [24, 25, 26]; see also details below.

While the above ideas have been developed in the linear realm, a parallel track of efforts has concerned a wide range of nonlinear models bearing localized modes, as summarized in a number of reviews [28]. The corresponding applications of the above studies are broad and far-reaching and extend from the nonlinear optics of waveguide arrays [29] to the dynamics of atomic Bose-Einstein condensates in periodic potentials [30] and from the micro-mechanical models of cantilever arrays [31] and electrical lattices [32] to prototypical models of the DNA double strand [33]. Two of the most canonical mathematical sets of models where localized nonlinear waveforms emerge consist of the discrete nonlinear Schrödinger (DNLS) model [34] and the nonlinear discrete variant of the wave equation in the form of nonlinear Klein-Gordon lattices [28, 35]. These represent generic frameworks featuring the interplay of discreteness, dispersion and nonlinearity and are in one or another variant (i.e., for different nonlinearities, external potentials etc.) involved in the description of each one of the above physical settings.

In this work we combine these two aspects of dynamical lattices. More specifically, we study the problem of constructing an embedded mode (EM) inside the band of three different, but related systems: (a) A nonlinear DNLS lattice (section II) (b) A $\phi^{4}$ chain (section III) (c) A one-dimensional bi-inductive electrical lattice (section IV). Our aim is, upon construction of the linear EM mode, to continue the relevant mode in the nonlinear realm, either as a solitary wave (in the DNLS setting) or as a discrete breather (in the $\phi^{4}$ and electrical lattice cases) and continue it parametrically over frequencies. We observe that the relevant modes can be continued within the (discretized —due to the finiteness of the domain—) continuous spectrum, yet their stability changes along this continuation. We detail the relevant instabilities and the corresponding bifurcation diagrams. We also dynamically monitor the instability evolution, as well as propose an experimentally realistic setting for the implementation of the ideas proposed herein. We believe that the relevant findings offer a sense of the potential applicability of EMs in nonlinear dynamical lattices that are of wide relevance in the above discussed physical settings. Our presentation closes with section V which summarizes our findings and presents our conclusions, as well as some interesting directions for further study. We also note in passing that the EMs obtained here are distinct from earlier works involving the discretization of a continuous model that, in turn, bears embedded solitons and where the EMs are genuinely nonlinear entities; an example of the latter in a (spatially homogeneous, i.e., involving no external potential) lattice model with competing quadratic and cubic nonlinearities has been studied in ref.[36].

We start by considering a discrete model described by the DNLS equation with a set of impurities following the ideas of [24, 25, 26] in the form:

where $n=1\dots N$, while $N$ represents the size of the system. We consider fixed-end boundary conditions, and $\epsilon_{n}$ represents the effective external potential leading to the existence of the EM. As is customarily the case in such DNLS lattices [34], the energy $H=\sum(C/2)|\Psi_{n+1}-\Psi_{n}|^{2}-(\gamma/2)|\Psi_{n}|^{4}+\epsilon_{n}|\Psi|^{2}$ and the power $P=\sum|\Psi_{n}|^{2}$ are conserved magnitudes. Stationary states in such DNLS settings arise in the form of standing waves: $\Psi_{n}=\phi_{n}\exp(i\omega t)$, with $\phi_{n}$ being the solutions of the steady state problem:

and $\omega$ are the corresponding frequencies.

We now turn to a $\phi^{4}$ Klein-Gordon chain described by the equations

supplemented by free-end boundary conditions. Here $n=1\dots N$, $N$ is the size of the system. $C$ is the coupling parameter, $\omega_{0}$ the frequency of the system in the linear limit and $s$ the nonlinear parameter that can be positive (hard potential) or negative (soft potential) [28]. We consider free boundary conditions, and $\epsilon_{n}$ prescribes the external potential leading at the linear limit to an EM, and has the same spatial profile as before. This system is Hamiltonian with an energy given by

This is the sole conserved quantity of this nonlinear dynamical lattice. Both for this reason, but also because the dependence of $H$ on the frequency $\omega$ of the breathers has been identified as a key indicator for their stability [38], we select to show in the relevant bifurcation diagrams below the dependence of $H$ on $\omega$.

In the homogeneous chain ($\epsilon_{n}=0$ $\forall n$), it is possible for a set of control parameter values to have stable nonlinear excitations (discrete breathers) with frequencies above the linear frequency band (phonon band) in the hard potential case (of $s=1$) and below this band in the soft case (of $s=-1$) [28]. Once again, the ideas of [24, 25, 26] enable the construction of an EM at the linear limit, while our aim here is to explore, upon “sculpting” an appropriate external potential profile $\epsilon_{n}$, the possibility of the existence, within the nonlinear problem, of spatially localized states in the continuous spectrum band with frequencies at or close to the corresponding linear mode. These discrete breathers should, naturally, be reduced to spatially localized linear excitations embedded in the continuous spectrum in the linear limit. An illustration of the latter is shown in Fig. 10, as constructed by analogy to what was reported previously for the DNLS (the two linear problems are qualitatively identical).

The realm of electrical lattices has a time-honored history as a highly controllable experimental testbed for the consideration of localized modes [40, 41], as well as numerous variants thereof, such as traveling [42], higher-dimensional realizations [43], connections with the Kuramoto model [44], and considerations of impurities [45], among many others. Here, we propose an electric line composed by resonant elements ($LC$ oscillators) coupled by inductors as shown in Fig. 14. This system is rather analogous to the ones used for the above mentioned experimental setups and can be tailored to the needed properties. To induce the excitation we need an external voltage source and a load resistor. The load resistor and the external voltage source will, with the right phase, allow us to induce the localized state. Once the localized state takes place we will turn off the driving and the load resistance. Thus, we could observe, for a suitable time interval (given the dissipative effects), this state.

For example, in line with with parameter values that have been used in some of the above experimental works, we will consider values of capacitance $C=770$ pF, inductances $L_{1}=680\mu$H, $L_{2}=330\mu$H, resistance $R=10$k$\Omega$ and voltage $V_{s}(t)=V_{d}\cos(\omega t+\varphi_{n}).$ A chain size that is quite suitable for experimental observations, in line with the above works, is $N=31$ with free boundary conditions. The (dimensionless) linear modes are the solutions of the equation

hence this creates the possibility to induce a spatially localized linear mode by using a set of impurities $\{\epsilon_{n}\}$, as an effective potential enabling the existence of an EM. To build the relevant circuit we have to replace inductors $L_{2}$ with a set of inductors $L_{2}^{(n)}$ whose values depend on the site, so as to create the effective local potential. Thus, in this case:

In figure 15 we show different values of inductors to obtain a linear localized state as shown in Fig. 16. By replacing the capacitor by a varactor it is possible to explore nonlinear localized modes in the phonon band. In principle, we believe that this concrete proposal renders the phenomenology presented herein accessible to the current state-of-the-art in electrical lattice experiments. This is true not only at the linear level described above, but also at the nonlinear one for suitably larger values of the voltage.

In the present work, we have explored the possibility of realizing embedded modes in the continuous spectrum of nonlinear dynamical lattices. Leveraging earlier ideas from the realm of linear problems, we have systematically constructed such modes in the linear (low-intensity) limit of the nonlinear problem. Then, the use of nonlinearity (focusing or defocusing, soft or hard) has allowed us to continue the relevant linear state in the presence of nonlinearity at arbitrary strengths detuning away from the linear limit. We have found that such states can arise as nonlinear extensions both at the solitary wave setting of the DNLS model and in the discrete breather realm of nonlinear Klein-Gordon dynamical lattices. Moreover, in the vicinity of the linear limit the states may be stable, while they destabilize, typically towards spreading their mass throughout the lattice, further away from the limit. Not only have we explored the existence, stability and dynamics of such EM states, but we have also proposed a concrete experimental implementation thereof in the form of a 31-node electrical lattice of inductors and capacitors (and resistors, and external drive, as is typical in such experiments).

It will of course be interesting to attempt to pursue such lattices in future experimental work and to attempt to explore the damped-driven variants of the relevant models (as in such experiments it is not straightforward to completely eliminate dissipation towards the Hamiltonian realm as in this study).

Another interesting direction is to examine generalizations of such modes in higher dimensions and the exploration of how higher-dimensionality may affect the stability conclusions reached herein. As yet another interesting point, considering such EMs close to the band edge of the continuous spectrum and examining how their stability may be affected by this effective “distance” in frequency from the band edge would be another interesting property to explore in the context of such nonlinear embedded modes. These topics are currently under investigation and associated findings will be reported in future publications.

M. I. M. acknowledges support by FONDECYT Grant 1200120. J.C-M. thanks the Regional Government of Andalusia under the project P18-RT-3480 and MICINN, AEI and EU (FEDER program) under the project PID2019-110430GB-C21. F.P. acknowledges financial support from the Ministerio de Ciencia, Innovación y Universidades (MICIU, Spain) through Project No. PID2019-108508GB-I00 cofinancied by FEDER funds. This material is based upon work supported by the US National Science Foundation under Grant DMS-1809074 (PGK).