Dynamical barrier for the formation of solitary waves in discrete lattices
Introduction
In the past few years, there has been an explosion of interest in discrete models that has been summarized in numerous recent reviews [1]. This growth has been spurted by numerous applications of dynamical lattice nonlinear models in areas as diverse as the nonlinear optics of waveguide arrays [2], the dynamics of Bose–Einstein condensates in periodic potentials [3], micro-mechanical models of cantilever arrays [4], or even simple models of the complex dynamics of the DNA double strand [5]. Perhaps the most prototypical model among those that emerge in these settings is the, so-called, discrete nonlinear Schrödinger equation (DNLS) [6]. DNLS may arise as a direct model, as a tight binding approximation, or even as an envelope wave expansion: it is, arguably, one of the most ubiquitous models in the nonlinear physics of dispersive, discrete systems.
In at least one of these settings (namely, in the nonlinear optics of waveguide arrays with the focusing nonlinearity), the feature that will be of interest to the present work has been observed experimentally. In particular, it has been noted, to the best of our knowledge firstly in Ref. [7], that when an injected beam of light into one waveguide had low intensity, then the beam dispersed through quasi-linear propagation. On the other hand, in the same work, experiments with high intensity of the input beam led to the first example of formation of discrete solitary waves in waveguide arrays. A very similar “crossover” from linear to nonlinear behavior was also observed very recently in arrays of waveguides with the defocusing nonlinearity [8]. The common feature of both works is that they used the DNLS equation as the supporting model to illustrate this behavior at a theoretical/numerical level. However, this crossover phenomenon is certainly not purely discrete in nature. Perhaps the most famous example of a nonlinear wave equation that possesses such a threshold is the integrable continuum nonlinear Schrödinger equation [9]. Specifically, it is well known that, e.g., in the case of a square barrier of initial conditions of amplitude and width L, the product determines the nature of the resulting soliton, and if it is sufficiently small the initial condition disperses without the formation of a solitonic structure [10]. On the other hand, the existence of the threshold is not a purely one-dimensional feature either. For instance, experiments on the formation of solitary waves in two-dimensional photorefractive crystals show that low intensities lead to diffraction, whereas higher intensities induce localization [11], [12]. Moreover, similar phenomena were observed even in the formation of higher-order excited structures such as vortices (as can be inferred by carefully inspecting the results of Refs. [13], [14]). It should be mentioned that the latter field of light propagation in photorefractive crystals is another major direction of current research in nonlinear optics; see, e.g., Ref. [15] for a recent review.
This crossover behavior between linear and nonlinear dynamics may be understood qualitatively rather simply. In the case of power law nonlinearities of order p, which are relevant in these settings, a small intensity , where , yields a nonlinear contribution that is negligible with respect to the linear terms of the equation. On the other hand, if , the opposite will be true and the nonlinear terms will dominate the linear ones, yielding essentially nonlinear behavior. A key question regards the details of this crossover and what determines its more precise location for an appropriately parametrized initial condition. This is the question we address herein. We argue that the problem related to the experiments described above reduces, at the mathematical level, to a DNLS equation with a Kronecker-δ initial condition parametrized by its amplitude. Then, a well defined value of the initial-state amplitude exists such that initial states with higher amplitude always give rise to localized modes. The condition may be determined by comparing the energy of the initial state with the energy of the localized excitations that the model supports. This sufficient, but not necessary, condition for the formation of localized solitary waves provides an intuitively and physically appealing interpretation of the dynamics that is in very good agreement with our numerical observations. We also consider variants of this process in different settings: for reasons of completeness, we present it also in the continuum NLS equation, noting the significant differences that the latter case has from the present one. As yet another example of very different (from both its non-integrable sibling and its continuum limit) dynamical behavior, we also present the case of the integrable discrete NLS (so-called Ablowitz–Ladik [16], [17], [18]) model. In addition to the one-dimensional DNLS lattice, we also consider the two-dimensional case where the role of both energy and beam power (mathematically the squared -norm) become apparent. We should note here that our tool of choice for visualizing the “relaxational process” (albeit in a Hamiltonian system) of the initial condition will be energy-power diagrams. Such diagrams have proven very helpful in visualizing the dynamics of initial conditions in a diverse host of nonlinear wave equations. In particular, they have been used in the nonlinear homogeneous systems such as birefringent media and nonlinear couplers as is discussed in Chapters 7 and 8 of Ref. [19]. They have also been used in a form closely related to the present work (but in the continuum case; see also the discussion below) for general nonlinearities in dispersive wave equations in [20], while they have been used to examine the migration of localized excitations in DNLS equations in [21].
Our presentation is structured as follows. In Section 2, we present the analytically tractable theory of the integrable “relatives” of the present model: we review the known theory for the NLS model and develop its analog for the integrable discrete NLS case. Then, in Section 3, we present our analytical and numerical results in the one- and two-dimensional DNLS equation. In the last section, we summarize our findings and present our conclusions, as well as highlight some important questions for future studies.
Section snippets
The continuum NLS model
For reasons of completeness of the presentation and to compare and contrast the results of the non-integrable case that is at the focus of the present work, we start by summarizing the threshold conditions for the continuum NLS model [10]. For the focusing NLS equation with squared barrier initial data (the inverse of) the transmission coefficient, , which is the first entry of the scattering matrix, is given by
Threshold conditions for the non-integrable DNLS model
We now turn to the non-integrable DNLS lattice that has the general form: where is a complex field (corresponding to the envelope of the electric field in optics, or the mean field wavefunction in optical lattice wells in the BECs), is the discrete Laplacian, and ϵ is the ratio of the tunneling strength to the nonlinearity strength. Using a scaling invariance of the equation, we can scale , through rescaling and . Importantly for our
Conclusions and future challenges
The above study has examined the presence of a sharp crossover between the linear and nonlinear dynamics of a prototypical dynamical lattice model such as the discrete nonlinear Schrödinger equation. This crossover has already been observed in media with the focusing nonlinearity [7] (as considered here). It has also been observed very recently in media with the defocusing nonlinearity [8]. The latter can be transformed into the former under the so-called staggering transformation ,
Acknowledgements
Y.D. gratefully acknowledges discussions with L. Isella. P.G.K. gratefully acknowledges support from grants NSF-DMS-0505663, NSF-DMS-0619492 and NSF-CAREER.
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