Bifurcation structures and dominant modes near relative equilibria in the one-dimensional discrete nonlinear Schrödinger equation

Abstract

We investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear Schrödinger equation (DNLSE) when the period of these orbits and a suitable defect act as bifurcation parameters. We find a reduced Hamiltonian that gives substantial insight into the dynamics of this system. The convexity of this Hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions. We show that the local loss of convexity in the reduced Hamiltonian is determined by the Hessian of its integrable part in the family of relative equilibria under study. Stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions, whose origin is suggested by the averaging principle. We also show that the return times form an optimal embedding to characterize the system dynamics. We show that the power spectral density of a suitable interference signal, arising from a ring of Bose–Einstein condensates and described by the DNLSE, has a single prominent peak at the breather-like relative equilibria.

Introduction

An area of increasing scientific interest over the past decades is the study of nonlinear effects in one-dimensional lattices. A classical theoretical model within this category is the discrete nonlinear Schrödinger equation (DNLSE) [1]. The interest in this model was driven by its usefulness in describing a large class of discrete nonlinear systems such as optical fibers [2], [3], [4], polarons [5], small molecules such as benzene [5], [6], and, more recently, dilute Bose–Einstein condensates (BEC) trapped in a multiwell periodic potential [7], [8], [9], [10]. Recent proposals to study rings of BEC in multiwell potentials, as in our model, have been reported using the transverse interference patterns of laser beams [11], [12]. Moreover, experimental studies of BEC have greatly benefited from developments in atom chip technology [13]. In these microchips, BEC systems have been created and manipulated efficiently [13].

Among the main effects in these lattices are the localized excitations in perfectly periodic strongly nonlinear systems [1], [14], [15]. We refer to discrete breathers, also known as intrinsic localized modes [1]. These are spatially localized, time-periodic and stable solutions of the DNLSE [1], and have been observed in many physical systems [15]. The evolution of the aforementioned BEC is governed by the Gross–Pitaevskii equation and can be reduced in the tight-binding approximation to a DNLSE [7], [8]. The existence of breathers in infinite one-dimensional lattices has been proved by MacKay and Aubry [16], while that of quasiperiodic breathers was shown by Bambusi and Vella [17].

The DNLSE has many relevant issues such as the dynamics of discrete breathers in one- or two-dimensional infinite lattices [1], propagation of excitations in the presence of disorder [18], mobility and interaction of breathers [14], [19]. We concentrate on the topic of the bifurcation structure of the relative equilibria and the dynamics of nearby quasiperiodic solutions. In this system the period of the relative equilibria and a suitable defect induced bifurcations when the number of oscillators in a ring geometry is small. By relative equilibria we mean orbits that are at equilibria in a suitable co-rotating frame; in other words, the trajectory is contained in a single symmetry group orbit [20]. The DNLSE has the $\mathrm{SO(2)}$ symmetry group as a result of the angular momentum conservation law.

The DNLSE is a nonlinear Hamiltonian system with $M$ degrees-of-freedom, where $M$ stands for the number of multiwell periodic potentials in a BEC [7], [8], [10] or the number of waveguides in an optical array [2], [3], [4]. The DNLSE, as is well known, has two constants of motion [1]. Therefore, when $M=2$, the DNLSE is integrable. However, when $M\ge 3$, the DNLSE can exhibit an amazing degree of complexity [21], [22], [23]. Indeed, the resonant behavior in three-degrees-of-freedom or higher-degrees-of-freedom Hamiltonian systems is an area of increasing scientific and technical interest [24]. In contrast to previous studies of the DNLSE [21], [22], [23], we consider a reduced Hamiltonian in action-angle variables. In recent articles [25], [26], [27] nontrivial features absent in the infinite lattice limits, such as the coexistence of linearly stable relative equilibria, were found for suitable parameters in rings with small $M$. In our previous work [25], [28] we discussed the main types of behavior in a ring with seven oscillators in the neighborhood of a linearly stable breather-like, but relatively easily destabilized, solution. The latter belongs to a family of asymmetric solutions, where strongly stable breather-like solutions are located. In our current study we focus on the global bifurcation structure of a family of relative equilibria and the nature of the quasiperiodic solutions in a neighborhood of that breather-like solution, where a single one-site defect induces interesting phenomena. The number of oscillators considered in this paper, $M=7$, allows for the manifestation of complex chaotic phenomena [25], [28]. However, we also find robust quasiperiodic dynamics, which is the main subject of the current paper, when suitable conditions are met. In particular, we investigate what occurs when a single one-site defect is present in the neighborhood of breather-like solutions [25], [28]. In [25], [28] we found quasiperiodic dynamics, localized chaos and delocalized chaos. In the latter case, we observed a well-approximated symbolic synchronization of information for three signals, which account for suitable population differences in a BEC. For $M<7$, the number of these signals is smaller than three. This makes the case $M=7$ special for our study. We have also considered the cases $M=9$ and $M=11$, where the bifurcations become increasingly more complicated as the on-site defect changes.

Our current study also considers a reduced Hamiltonian (RH), where the DNLSE relative equilibria become equilibria for the RH. Moreover, by showing the quasi-integrability of the RH near its linearly stable equilibria, we address stability considerations of nearby quasiperiodic solutions. Our study reveals that the local convexity properties of the integrable part of the RH are sufficient conditions to predict Lyapunov stability in a family of RH equilibria. The latter stability considerations allow us to predict the presence of families of periodic orbits. The quasiperiodic solutions are also studied by means of a suitable Poincaré surface of section. We mainly use criteria based on the power spectral density and embedding techniques to show the presence of a few dominant modes. The analysis is simplified by considering a set of slow and fast variables, which are predicted by the averaging principle [29], [30].

This article has six sections. The DNLSE model is discussed in Section 2. In Section 3, we present a bifurcation structure of a family of relative equilibria of this model. The reduced Hamiltonian (RH) and its properties along different families of relative equilibria is considered in Section 4. In Section 5, using different approaches, we identify tori which display dominant modes. As an application of our results, we discuss a possible experiment in a BEC system. Finally, in Section 6, we summarize our main conclusions.