Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

Abstract

We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interactionand with mass m. Using the exact analytic form for rest frame solitary waves of the form (x,t) = ψ(x)e −iωt for arbitrary κ, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed fourth-order operator splitting integration method. For different ranges of κ we map out the stability regimes in ω. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time t c , it takes for the instability to set in, is an exponentially increasing function of ω and t c decreases monotonically with increasing κ.

Nonlinear Dirac equation solitary waves under a spinor force with different components

“…Recently, for the Soler model, we found that all stable NLD solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable [41]. This result is consistent with the rigorous analysis in the nonrelativistic limit [42].…”

“…For v = 0, the soliton travels only a short distance, therefore a final integration time of t f = 3000 can be taken, which is technically the maximum time in our simulation program. The computational cost taking the maximum time is huge because our fourth-order operator splitting method that we have used earlier [41] [53] requires that the spatial spacing h = τ 12 , where we choose the time step τ = 0.025. This implies that the number of grid points is 96, 000 for the above system size.…”

Solitary waves in a discrete nonlinear Dirac equation

“…In the cubic case (k = 1) it is predicted in [17] that Λ c = 0.6976. However, in a recent paper [16], further numerical simulations have suggested that solitons may be dynamically stable for Λ ≥ 0.56.…”

“…Various aspects have been examined in this context, including the spectral stability and the potential emergence of point spectrum eigenvalues with nonzero real part (which has been shown to be impossible to happen beyond the so-called embedded thresholds) [11], the orbital and asymptotic stability under a series of relevant assumptions [12], the nonlinear Schrödinger (non-relativistic) limit and its instability for nonlinearities beyond a critical exponent [13], as well as classical (Vakhitov-Kolokolov) and more suitable to this setting (energy based) criteria [14] for the linear stability of solitary waves in the NLDE. A series of more computationally/physically oriented studies both in the context of the stability/dynamics of the NLDE solitary waves [15,16] (again, in principle for arbitrary nonlinearity powers) and in that of these structures in the presence of external fields [17] have also recently appeared.…”

Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation

“…Some analytical solitary wave solutions have been obtained in a series of research works [8][9][10]. With regard to more general initial conditions, however, it is difficult to derive the exact solutions of the NLD equation.…”