On spectral stability of the nonlinear Dirac equation

Abstract

We study the point spectrum of the nonlinear Dirac equation in any spatial dimension, linearized at one of the solitary wave solutions. We prove that, in any dimension, the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds. We then prove that the birth of point eigenvalues with nonzero real part (the ones which lead to linear instability) from the essential spectrum is only possible from the embedded eigenvalues or thresholds, and therefore can not take place beyond the embedded thresholds. We also prove that "in the nonrelativistic limit" ω → m, the point eigenvalues can only accumulate to 0 and ±2mi.Given a real-valued function f ∈ C(R), f (0) = 0, we consider the following nonlinear Dirac equation in R n , n ≥ 1, which is known as the Soler model [Sol70]:Above, D m = −iα · ∇ + βm is the free Dirac operator. Here α = (α  ) 1≤≤n , with α  and β the self-adjoint N × N Dirac matrices (see Section 1.1 for the details); m > 0 is the mass. We are interested in the stability properties of the solitary wave solutions to (1.1):where the amplitude φ ω satisfies the stationary equationIn the present work, we study the spectral stability of solitary waves in the nonlinear Dirac equation. Given a particular solitary wave (1.2), we consider its perturbation, φ ω (x) + ρ(x, t) e −iωt , and study the spectrum of the linearized equation on ρ.Definition 1.1. We will say that a particular solitary wave is spectrally stable if the spectrum of the equation linearized at this wave does not contain points with positive real part.Remark 1.2. Sometimes it is convenient to include a further requirement that the operator corresponding to the linearization at this wave does not contain 4 × 4 Jordan blocks at λ = 0 and no 2 × 2 Jordan blocks at λ ∈ iR \ {0}; such blocks are expected to lead to dynamic instability. Related definitions of linear instability are in [Cuc14,BC12b].In the context of spectral stability, the well-posedness of the initial value problem associated with (1.1) is not crucial. However in the investigation of the dynamical or Lyapunov stability the local well-posedness would be essential. Once again the complete account of the works on this subject is beyond our objectives but we briefly mention some of the classical results related to the three-dimensional case. Escobedo and Vega [EV97] proved the local wellposedness in the H s -setting with s > (n − 1)/2. The charge-critical scaling power (at least in the massless case) being (n − 1)/2, some works have been devoted to reach this endpoint. For instance, Machihara, Nakamura, Nakanishi and Ozawa show in [MNNO05] that the H 1 regularity in the radial variable with an arbitrarily small regularity in the angular one is sufficient. This for instance settles the H 1 -well-posedness problem for radially symmetric initial data. This can for instance solve the problem for initial data of the form (4.10) for (1.1) in dimension 3. For (1.1), the non-linearity presents some null-structure which was exploited by B...

Vakhitov–Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields

“…In the nonrelativistic limit ω m, for k ∈ (0, 2), one has spectral stability according to [BC12b]; for k > 2, there is linear instability by [CGG14].…”

“…In view of recent results on stability and instability for the nonlinear Dirac equation [CKMS10,BC12a,BC12b,CGG14] it is becoming clear that the VK criterion is still useful for the spinor systems in the nonrelativistic limit, when the amplitude of solitary waves is small. In particular, the ground states ("smallest energy solitary waves") in the charge-subcritical nonlinear Dirac equation (with the nonlinearity of order 2k + 1, with k < 2/n) are linearly stable in the nonrelativistic limit ω m, which corresponds to solitary waves of small amplitudes.…”

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

“…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].…”

Solitary Waves of a <inline-formula> <tex-math notation="LaTeX">$\mathcal {P}$</tex-math> </inline-formula><inline-formula> <tex-math notation="LaTeX">$\mathcal {T}$</tex-math> </inline-formula>-Symmetric Nonlinear Dirac Equation

“…2. For the rest of the spectrum we note that according to [25], eigenfrequencies with nonzero imaginary part can only be born in the interval − 1 − |Λ|, 1 + |Λ| . Here, however, all the eigenfrequencies remains inside this interval for all γ, as illustrated in Fig.…”

“…While such models were proposed in the context of high-energy physics over 50 years ago [11], [12], they have, arguably, been far less widespread than their nonrelativistic counterpart [13], the nonlinear Schrödinger (NLS) equation [14], [15]. In recent years, however, there has been a surge of activity around NLDE models fueled to some extent by analytical solutions and computational issues arising in associated numerical simulations [16]- [19], as well as by the considerable progress achieved by rigorous techniques towards aspects of the spectral, orbital and asymptotic stability of solitary wave solutions of such models [20]- [25] and towards criteria for their spectral stability [26], [27]. Although our emphasis herein will be on the so-called Gross-Neveu model [28] (sometimes also referred to as the Soler model [29]), we also mention in passing that another main stream of activity in this direction has been towards the derivation of NLDEs in the context e.g.…”