We consider Hamiltonian systems with U (1) symmetry. We prove that in the generic situation the standing wave that has the minimal energy among all other standing waves is unstable, in spite of the absence of linear instability. Essentially, the instability is caused by higher algebraic degeneracy of the zero eigenvalue in the spectrum of the linearized system. We apply our theory to the nonlinear Schrödinger equation.

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.

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...

Existence and stability of Dirac points in the dispersion relation of operators periodic with respect to the hexagonal lattice is investigated for different sets of additional symmetries. The following symmetries are considered: rotation by 2π/3 and inversion, rotation by 2π/3 and horizontal reflection, inversion or reflection with weakly broken rotation symmetry, and the case where no Dirac points arise: rotation by 2π/3 and vertical reflection.All proofs are based on symmetry considerations. In particular, existence of degeneracies in the spectrum is deduced from the (co)representation of the relevant symmetry group. The conical shape of the dispersion relation is obtained from its invariance under rotation by 2π/3. Persistence of conical points when the rotation symmetry is weakly broken is proved using a geometric phase in one case and parity of the eigenfunctions in the other.