Stability of Solitary Waves and Vortices in a 2D Nonlinear Dirac Model

Abstract

We explore a prototypical two-dimensional massive model of the nonlinear Dirac type and examine its solitary wave and vortex solutions. In addition to identifying the stationary states, we provide a systematic spectral stability analysis, illustrating the potential of spinor solutions to be neutrally stable in a wide parametric interval of frequencies. Solutions of higher vorticity are generically unstable and split into lower charge vortices in a way that preserves the total vorticity. These conclusions are found not to be restricted to the case of cubic two-dimensional nonlinearities but are found to be extended to the case of quintic nonlinearity, as well as to that of three spatial dimensions. Our results also reveal nontrivial differences with respect to the better understood nonrelativistic analogue of the model, namely the nonlinear Schrödinger equation. DOI: 10.1103/PhysRevLett.116.214101 Introduction.-In the context of dispersive nonlinear wave equations, admittedly the prototypical model that has attracted a wide range of attention in optics, atomic physics, fluid mechanics, condensed matter, and mathematical physics is the nonlinear Schrödinger equation (NLS) [1][2][3][4][5][6][7]. By comparison, far less attention has been paid to its relativistic analogue, the nonlinear Dirac equation (NLD) [8], despite its presence for almost 80 years in the context of high-energy physics [9][10][11][12][13]. This trend is slowly starting to change, arguably, for three principal reasons. Firstly, significant steps have been taken in the nonlinear analysis of stability of such models [14][15][16][17][18][19], especially in the one-dimensional setting. Secondly, computational advances have enabled a better understanding of the associated solutions and their dynamics [20][21][22][23][24]. Thirdly, and perhaps most importantly, NLD starts emerging in physical systems that arise in a diverse set of contexts of considerable interest. These contexts include, in particular, bosonic evolution in honeycomb lattices [25,26] and a growing class of atomically thin 2D Dirac materials [27], such as graphene, silicene, germanene, and transition metal dichalcogenides [28] (notice that in this Letter, we use nD to refer to n spatial dimensions). Recently, the physical aspects of nonlinear optics, such as light propagation in honeycomb photorefractive lattices (the socalled photonic graphene) [29,30], have prompted the consideration of intriguing dynamical features, e.g., conical diffraction in 2D honeycomb lattices [31]. Inclusion of nonlinearity is then quite natural in these models, although in a number of them (e.g., in atomic and optical physics) the nonlinearity does not couple the spinor components.

Conical intersections for light and matter waves

“…It is therefore of interest to fabricate photonic media combining strong optical nonlinearities with conical intersections to enable the study of both all-optical beam control, and fundamental effects such as the existence and stability of solitons [20,87,88].…”

Ring Dirac solitons in nonlinear topological systems

“…Recently, a stable 2D soliton of the nonlinear Dirac equation has been found by Cuevas-Maraver et al [37,38]. The main result of Ref.…”

“…In what follows we focus on the solitons with zero angular momentum, m = 0, since the high-momentum states were shown to be unstable [37]. In order to solve Eqs.…”

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

“…The time evolutions of the charge density from 푡 = 0 to 푡 = 30 are shown in Figure 1. The collision phenomenon of two solitary waves is depicted in Figure 1, from which we can see that the solitary waves keep moving with their original velocities and shapes after the collision [34][35][36].…”