On Spectral Stability of Solitary Waves of Nonlinear Dirac Equation in 1D

Abstract

Abstract.We study the spectral stability of solitary wave solutions to the nonlinear Dirac equation in one dimension. We focus on the Dirac equation with cubic nonlinearity, known as the Soler model in (1+1) dimensions and also as the massive Gross-Neveu model. Presented numerical computations of the spectrum of linearization at a solitary wave show that the solitary waves are spectrally stable. We corroborate our results by finding explicit expressions for several of the eigenfunctions. Some of the analytic results hold for the nonlinear Dirac equation with generic nonlinearity.

On linear instability of solitary waves for the nonlinear Dirac equation

“…In spite of this, the question of spectral stability of solitary waves of nonlinear Dirac equation is still completely open. Numerical results [BC12a] show that in the 1D Soler model (cubic nonlinearity) all solitary waves are spectrally stable. We also mention the related numerical results in [CP06,Chu08].…”

“…We first give a simple demonstration of the existence and uniqueness of solitary waves in one dimension, following the article [BC12a], and allowing for more general nonlinearities f (s). …”

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

“…This is present in the spectrum of the linearized equation due to the U(1) symmetry and due to the translational symmetry, which are both preserved when γ = 0, hence both the algebraic and geometric multiplicity of this eigenvalue are preserved for all values of γ, as is the presence of two generalized eigenvectors. The spectrum also features the eigenfrequencies ω = ±2Λ [19] which can not leave the imaginary axis since their presence in the spectrum is due to the SU (1,1) invariance (see the more detailed discussion below), which is also preserved for any γ; the relevant eigenfrequency, which persists under variations of γ, can be discerned in the left panel of Fig. 2.…”

“…Remarkably, such solitary waves with oscillating charge can be attained by performing an SU(1,1) transformation to a standing wave soliton (2) and have been predicted in [36]. This type of solution for a standing wave of frequencyΛ is, in fact, intrinsically connected to the invariance of the frequency 2Λ in the spectrum [19]. More specifically, these bi-frequency, oscillating charge coherent structures [which can be dubbed as SU (1,1) solitons] are of the form:…”

“…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.…”