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We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction, known as the Soler model (with an arbitrary nonlinearity and in arbitrary dimension) and the Dirac-Klein-Gordon with Yukawa self-interaction. These solitary waves provide a natural implementation of qubit and qudit states in the theory of quantum computing. We show the relation of \begin{document}$± 2ω\mathrm{i}$\end{document} eigenvalues of the linearization at a solitary wave, Bogoliubov \begin{document}$\mathbf{SU}(1,1)$\end{document} symmetry, and the existence of bi-frequency solitary waves. We show that the spectral stability of these waves reduces to spectral stability of usual (one-frequency) solitary waves.

We study the point spectrum of the linearization at a solitary wave solution $\phi_\omega(x)e^{-\mathrm{i}\omega t}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the nonlinear term given by $f(\psi^*\beta\psi)\beta\psi$ (known as the Soler model). We focus on the spectral stability, that is, the absence of eigenvalues with nonzero real part, in the non-relativistic limit $\omega\lesssim m$, in the case when $f\in C^1(\mathbb{R}\setminus\{0\})$, $f(\tau)=|\tau|^k+O(|\tau|^K)$ for $\tau\to 0$, with $04/n$. An important part of the stability analysis is the proof of the absence of bifurcations of nonzero-real-part eigenvalues from the embedded threshold points at $\pm 2m\mathrm{i}$. Our approach is based on constructing a new family of exact bi-frequency solitary wave solutions in the Soler model, using this family to determine the multiplicity of $\pm 2\omega\mathrm{i}$ eigenvalues of the linearized operator, and the analysis of the behaviour of "nonlinear eigenvalues" (characteristic roots of holomorphic operator-valued functions)., Comment: 55 pages

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 associ- ated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical per- spective and then complemented with numerical computations. Finally, the dynam- ics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr¨odinger equation. AEI/FEDER, European Union project MAT2006-79866-R

In this work, we provide two complementary perspectives for the (spectral) stability of solitary traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. One is as an eigenvalue problem for a stationary solution in a cotraveling frame, while the other is as a periodic orbit modulo shifts. We connect the eigenvalues of the former with the Floquet multipliers of the latter and using this formulation derive an energy-based spectral stability criterion. It states that a sufficient (but not necessary) condition for a change in the wave stability occurs when the functional dependence of the energy (Hamiltonian) H of the model on the wave velocity c changes its monotonicity. Moreover, near the critical velocity where the change of stability occurs, we provide an explicit leading-order computation of the unstable eigenvalues, based on the second derivative of the Hamiltonian H(c0) evaluated at the critical velocity c0. We corroborate this conclusion with a series of analytically and numerically tractable examples and discuss its parallels with a recent energy-based criterion for the stability of discrete breathers. Unión Europea MAT2016-79866-R (AEI / FEDER, UE)

In this work, we provide two complementary perspectives for the (spectral) stability of solitary traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. One is as an eigenvalue problem for a stationary solution in a cotraveling frame, while the other is as a periodic orbit modulo shifts. We connect the eigenvalues of the former with the Floquet multipliers of the latter and using this formulation derive an energy-based spectral stability criterion. It states that a sufficient (but not necessary) condition for a change in the wave stability occurs when the functional dependence of the energy (Hamiltonian) H of the model on the wave velocity c changes its monotonicity. Moreover, near the critical velocity where the change of stability occurs, we provide an explicit leading-order computation of the unstable eigenvalues, based on the second derivative of the Hamiltonian H(c0) evaluated at the critical velocity c0. We corroborate this conclusion with a series of analytically and numerically tractable examples and discuss its parallels with a recent energy-based criterion for the stability of discrete breathers. Unión Europea MAT2016-79866-R (AEI / FEDER, UE)

We use the perturbation theory to build solitary wave solutions $\phi_\omega(x)e^{-i\omega t}$ to the nonlinear Dirac equation in $\mathbb{R}^n$, $n\ge 1$, with the Soler-type nonlinear term $f(\bar\psi\psi)\beta\psi$, with $f(\tau)=|\tau|^k+o(|\tau|^k)$, $k>0$, which is continuous but not necessarily differentiable. We obtain the asymptotics of solitary waves in the nonrelativistic limit $\omega\lesssim m$; these asymptotics are important for the linear stability analysis of solitary wave solutions. We also show that in the case when the power of the nonlinearity is Schr\"odinger charge-critical, one has $Q'(\omega), Comment: Approximately 44 pages

We study the spectral stability of solitary wave solutions to the nonlinear Dirac e quation 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 eigenfunction s. Some of the analytic results hold for the nonlinear Dirac equation with generic nonlinearity.

We consider the nonlinear Dirac equation, also known as the Soler model: $i\p\sb t\psi=-i\alpha \cdot \nabla \psi+m \beta \psi-f(\psi\sp\ast \beta \psi) \beta \psi$, $\psi(x,t)\in\mathbb{C}^{N}$, $x\in\mathbb{R}^n$, $n\le 3$, $f\in C\sp 2(\R)$, where $\alpha_j$, $j = 1,...,n$, and $\beta$ are $N \times N$ Hermitian matrices which satisfy $\alpha_j^2=\beta^2=I_N$, $\alpha_j \beta+\beta \alpha_j=0$, $\alpha_j \alpha_k + \alpha_k \alpha_j =2 \delta_{jk} I_N$. We study the spectral stability of solitary wave solutions $\phi(x)e^{-i\omega t}$. We study the point spectrum of linearizations at solitary waves that bifurcate from NLS solitary waves in the limit $\omega\to m$, proving that if $k>2/n$, then one positive and one negative eigenvalue are present in the spectrum of the linearizations at these solitary waves with $\omega$ sufficiently close to $m$, so that these solitary waves are linearly unstable. The approach is based on applying the Rayleigh--Schroedinger perturbation theory to the nonrelativistic limit of the equation. The results are in formal agreement with the Vakhitov--Kolokolov stability criterion., Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1203.3859 (an earlier 1D version)

Background This paper critiques the model for non-verbal communication referred to as SOLER (which stands for: "Sit squarely"; "Open posture"; "Lean towards the other"; "Eye contact; "Relax"). It has been approximately thirty years since Egan (1975) introduced his acronym SOLER as an aid for teaching and learning about non-verbal communication. Aim There is evidence that the SOLER framework has been widely used in nurse education with little published critical appraisal. A new acronym that might be appropriate for non-verbal communication skills training and education is proposed and this is SURETY (which stands for "Sit at an angle"; "Uncross legs and arms"; "Relax"; "Eye contact"; "Touch"; "Your intuition"). The new model The proposed model advances the SOLER model by including the use of touch and the importance of individual intuition is emphasised. The model encourages student nurse educators to also think about therapeutic space when they teach skills of non-verbal communication.

Consider an abstract Hamiltonian system which is invariant under a group of operators. We continue to study the effect of the group invariance on the stability of solitary waves. Applications are given to bound states and traveling wave solutions of nonlinear wave equations.

The stability problem of the localized solutions for classical Dirac fields with scalar self-interactions is considered in the framework of the Shatah-Strauss formalism. We study the stability and instability under dilations and provide an application to the Soler model.