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.

This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier-Stokes equations, complexified in time. Specifically, for periodic boundary conditions on [0, L] 2 , and a force g ∈ D(A α−12 ), we show there is a fixed strip about the real time axis on which a uniform bound A α u < mανκ α 0 holds for each α ∈ N. Here ν is viscosity, κ 0 = 2π L , and mα is explicitly given in terms of g and α. We show that if any element in A is in D(A α ), then all of A is in D(A α ), and likewise with D(A α ) replaced by C ∞ (Ω). We demonstrate the universality of this "all for one, one for all" law on the union of a hierarchal set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a nonzero force by showing that if this is so, the force must be in a particular function class.

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schrödinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT -symmetric variant of the model.

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We discuss the stability properties of the solutions of the general nonlinear Schrödinger equation (NLSE) in 1+1 dimensions in an external potential derivable from a parity-time (PT ) symmetric superpotential W (x) that we considered earlier [1]. In particular we consider the nonlinear partial differential equation {i ∂t +∂ 2 x −V (x)+g|ψ(x, t)| 2κ } ψ(x, t) = 0, for arbitrary nonlinearity parameter κ, where g = ±1 and V is the well known Pöschl-Teller potential which we allow to be repulsive as well as attractive. Using energy landscape methods, linear stability analysis as well as a time dependent variational approximation, we derive consistent analytic results for the domains of instability of these new exact solutions as a function of the strength of the external potential and κ. For the repulsive potential (and g = +1) we show that there is a translational instability which can be understood in terms of the energy landscape as a function of a stretching parameter and a translation parameter being a saddle near the exact solution. In this case, numerical simulations show that if we start with the exact solution, the initial wave function breaks into two pieces traveling in opposite directions. If we explore the slightly perturbed solution situations, a 1% change in initial conditions can change significantly the details of how the wave function breaks into two separate pieces. For the attractive potential (and g = +1), changing the initial conditions by 1 % modifies the domain of stability only slightly. For the case of the attractive potential and negative g perturbed solutions merely oscillate with the oscillation frequencies predicted by the variational approximation.