Research paperUnstaggered-staggered solitons on one- and two-dimensional two-component discrete nonlinear Schrödinger lattices
Introduction
Discrete nonlinear Schrödinger (DNLS) equations provide models for a great variety of physical systems [1]. A well-known implementation of the basic DNLS equation is provided by arrays of transversely coupled optical waveguides, as predicted in [2] and realized experimentally, in various optical settings [3], [4], [5], [6]. A comprehensive review of nonlinear optics in discrete settings was given by Ref. [7]. Another realization of the DNLS equation in provided by Bose-Einstein condensates (BECs) loaded into deep optical-lattice potentials, which split the condensate into a chain of droplets trapped in local potential wells, which are tunnel-coupled across the potential barriers between them [8], [9]. In the tight-binding approximation, this setting is also described by the DNLS version of the Gross-Pitaevskii (GP) equation [10], [11], [12], [13], [14].
One-dimensional (1D) DNLS equations with self-attractive and self-repulsive on-site nonlinearity generate localized modes of unstaggered and staggered types, respectively. In the latter case, the on-site amplitudes alternate between adjacent sites of the lattice [1]. In the continuum limit, the unstaggered discrete solitons carry over into regular ones, while the staggered solitons correspond to gap solitons, which are supported by the combination of self-defocusing nonlinearity and spatially periodic potentials [15], [16], [17].
Many physical settings are modeled by systems of coupled DNLS equations. In optics, they apply to the bimodal propagation of light represented by orthogonal polarizations or different carrier wavelengths. In BEC, coupled GP equations describe binary condensates [18]. Usually, bimodal discrete solitons in two-component systems are considered with a single type of their structure in both components, either unstaggered or staggered, because the self-phase- and cross-phase-modulation (SPM and XPM) terms, acting in each component and coupled nonlinearly, are assumed to have identical signs [1]. Nevertheless, the opposite signs are also possible in BEC, where either of them may be switched by means of the Feshbach resonance [18], [19], [20], [21], [22]. Discrete solitons of the mixed type, built as complexes of unstaggered and staggered components, were introduced in Ref. [23], assuming opposite SPM and XPM signs. Earlier, single-component states of a mixed unstaggered-staggered type were investigated in the form of surface modes at an interface between different lattices [24], [25]. In continuum systems, counterparts of mixed modes are represented by semi-gap solitons, which are bound states of an ordinary soliton in one component and a gap soliton in the other [26].
The mixed modes reported in Ref. [23] are “symbiotic” ones, as each component in isolation may support solely ordinary unstaggered solitons. The results were obtained in an analytical form, using the variational approximations (VA), and verified by means of numerical methods. It was found that almost all the symbiotic solitons were predicted by the VA accurately, and were stable. Unstable solitons were found only close to boundary of their existence region, where the solitary modes have very broad envelopes, being poorly approximated by the VA.
Most works on DNLS systems concern symmetric on-site-centered fundamental solitons, which represent the ground state of the corresponding model [1], including the unstaggered-staggered solitons [23]. Furthermore, only fundamental solitons represent stationary states in the continuum NLS equation. However, DNLS models give rise to stationary excited states, which may be stable too. The simplest among them are antisymmetric states in the form of discrete twisted solitons [27], which have no counterparts in the continuum limit. Once unstaggered-staggered discrete solitons are possible in two-component DNLS systems, it is natural to introduce the twist in the latter setting too, with three different species of such discrete solitons possible, which are single-twisted, in either component—staggered or unstaggered one—or double-twisted, in both components. One relevant physical example of twisted 1D lattice solitons can be found in the study of crystalline acetanilide lattices (see, for instance, Figure 6(e) in [104]).
Discrete solitons on two-dimensional (2D) lattices have been studied in a variety of contexts, [1], [28]. Experimental literature highlights the existence of real solitons on 2D optically induced nonlinear photonic lattices [29], [30]. A natural extension of the earlier analysis, performed in the 1D setting [23], is to build two-component unstaggered-staggered complexes in 2D two-component DNLS systems, which may be realized physically in the same physical settings (optics and BEC) as mentioned above, provided that the corresponding waveguiding arrays are built, in the transverse plane, as 2D lattices, or the BEC is loaded into a deep 2D optical-lattice potential. In addition to 2D fundamental discrete solitons which may be naturally expected in the unstaggered-staggered system, one may also look for compound modes in which one or both components are represented by discrete vortex solitons [31], which is the 2D analogue of the 1D twisted solitons. Vortex solitons have previously been found in various optical setups [32], [33], [34], see also recent reviews [35], [36]. Experimental creation of discrete vortex solitons in self-focusing optically induced lattices was reported in Refs. [30], [37].
The remainder of this paper is organized according to the dimension of the lattice domain, with single- and double-twisted two-component unstaggered-staggered solitons on 1D lattices considered in Section 2. The analysis of unstaggered-staggered 2D discrete soliton complexes, consisting of fundamental soliton pairs, along with the more sophisticated fundamental-vortical and vortical-vortical ones, is reported in Section 3. For both dimensions of the lattice, we present the governing DNLS equations and the corresponding Lagrangian. Assuming a decay rate for the solitons’ tails as predicted by the linearization of the DNLS equations, we elaborate the variational approximation (VA) for each type of soliton. We then use the VA-produced predictions as an initial guess to obtain the corresponding states in a numerically exact form. This approach is useful, as without an appropriate input the numerical scheme may readily converge to zero or some non-physical state. Furthermore, for stationary states which are stable and symmetric between the components (fundamental-fundamental, twist-twist, or vortical-vortical), the agreement of VA with numerical findings is quite good, whereas in the case of asymmetric pairs of the components (one twisted or vortical, the other being fundamental) the agreement is less accurate. Starting with numerically exact stationary states, we then simulate their evolution in time to determine what states are stable or unstable. Concluding remarks are made in Section 4.
Section snippets
One-dimensional coupled unstaggered-staggered modes
In this section, we initiate the analysis by formulating VA, which has proved to be quite efficient in the studies of fundamental discrete solitons in diverse settings, as shown at heuristic [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65] and more rigorous [66], [67] levels. We introduced the DNLS equations and their Lagrangian in Section 2.1, elaborate the VA in Section 2.2,
Two-dimensional unstaggered-staggered lattices
In the higher-dimensional case, both theoretical [31] and experimental [29], [30] work demonstrate that a variety of dynamics are possible for the DNLS. Here, we extend the consideration of unstaggered-staggered modes to 2D lattices. As in the 1D case, we first present the dynamical system and its Lagrangian in Section 3.1. We then derive the VA in Section 3.2, and present representative stable stationary two-component solitons in Section 3.3.
Conclusions
Extending the analysis of the recently introduced system of nonlinearly coupled DNLS equations with unstaggered and staggered components (which requires opposite signs of the SPM and XPM nonlinearities—a situation possible in binary BEC), we have elaborated families of 1D discrete solitons with a single twisted or both twisted components, complementing the earlier work on fundamental soliton pairs on unstaggered-staggered lattices [23]. Analytical solutions for the discrete solitons are
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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2021, Chaos, Solitons and FractalsCitation Excerpt :Moreover, the existence of QDs in two-component 1D Bose-Hubbard chains has been demonstrated [50]. If the modulation of the optical lattice is deep enough, one can use the tight-binding approximation [74], which permits the condensate order parameter to be written as a sum of the wave functions localized in each well of the periodic potential, to reduce the Gross-Pitaevskii equations (GPEs) to a discrete form [75–79]. Discrete solitons and breathers with dilute BECs, which are governed by the discrete form of the GPEs, have been studied [75].