Relativistic Nonlinear Equations Research Articles

Superconducting quantum criticality and the anomalous scaling: A nonlinear relativistic equation

By using the Landau-Ginzburg-Wilson paradigm, we show that, near a quantum critical point (QCP), Cooper pairs at zero temperature would obey a nonlinear relativistic equation, where the imaginary time emerges as a novel dimension. This relativistic equation is applicable to certain superconductors at zero temperature for which the Faber-Pippard coherence length formula holds at and above the upper critical dimension. Here, we further show that the relativistic equation leads to a testable prediction in the vicinity of the QCP Tc=0, with Tc being the transition temperature. That is, for 2D overdoped (clean) superconducting films, when the parameter Tc/(c0νF) is lower than a characteristic scale, the Lorentz symmetry of relativistic equation arouses an anomalous scaling ξ0∝Tc−1.34, where ξ0 denotes the zero-temperature coherence length, νF denotes the Fermi velocity, and c0 denotes the Faber-Pippard coefficient. However, when the parameter Tc/(c0νF) is large enough, the Lorentz symmetry may be broken so that the Faber-Pippard scaling ξ0∝Tc−1 is restored.

Effect of dynamical non-neutrality on the modulational instability of laser propagating through hot magnetoplasma

In this work, modulational instability of a laser pulse in a hot magnetized plasma is investigated. Nonlinear relativistic equation, describing the amplitude evolution of a laser with finite longitudinal and transversal structure, is obtained. Taking into account the plasma dynamical non-neutrality caused by the ponderomotive force, the growth rate of the modulational instability is derived. Effect of the pulse length on the instability growth rate is investigated.

Nonlinear electromagnetic theory of low-frequency instability of a relativistic electron beam in plasma with strongly nonpotential plasma and beam waves

A substantially nonpotential nonlinear theory of instability of a straight high-current relativistic electron beam propagating in a plasma waveguide under the conditions of the collective stimulated Cherenkov effect is elaborated. In the most general statement of problem, relativistic nonlinear equations for the temporal dynamics of this instability are obtained. The case of a low-density plasma is considered when both the plasma and beam waves are essentially nonpotential. Under the assumption that weak resonance plasma-beam interaction and linearity of plasma electrons are present, the general equations are transformed (by expanding the electron trajectories and momenta) to cubically nonlinear relativistic equations, which describe nonlinear shifts in frequencies of the plasma and beam waves. Analytic solutions to these equations are obtained with allowance for the dependence of the electron mass on the electron velocity and for the nonpotentiality of the interacting waves. The results are in good agreement with the numerical solutions of the general nonlinear equations in a wide range of values for the parameters.

Nonlinear theory of the collective Cherenkov interaction of a dense relativistic electron beam with a dense plasma in a waveguide

A nonlinear theory of the instability of a straight relativistic dense electron beam in a plasma waveguide is derived for conditions of the stimulated collective Cherenkov effect. A study is made of a waveguide with a dense plasma such that the plasma wave excited by the beam during the instability can be escribed, with a good degree of accuracy, as a potential wave. General relativistic nonlinear equations are btained that describe the temporal dynamics of beam-plasma instabilities with allowance for plasma nonlinearity and the generation of harmonics of the initial perturbation. Under the assumption that the resonant interaction between the beam waves and the plasma waves is weak, the general equations are reduced to relativistic equations with cubic nonlinearities by using the method of expansion in small perturbations of the trajectories and momenta of the beam and plasma electrons. The reduced equations are solved analytically, the time scales on which the instability saturates are determined, and the nonlinear saturation amplitudes are obtained. A comparison between analytical solutions to the reduced equations and numerical solutions to the general nonlinear equations shows them to be in good agreement. Nonlinear processes caused by the relativistic nature of the beam are found to prevent stochastization of the system in the nonlinear stage of the well-developed instability. In contrast, a nonrelativistic electron beam is found to be subject to significant anomalous nonlinear stochastization.

Equations describing strong laser field in interaction with matter within a plasma in the hydrodynamic limit

General equations proving useful in the context of strong electron photon interactions are derived. Such interactions are driven by launching short and powerful laser pulses on a material medium. Relativistic nonlinear equations are obtained from the system of fluid and Maxwell equations through a suitable perturbative approach into three main parameters: q (energy parameter), δ (time parameter), and α (spatial parameter). Numerous phenomena are recovered—such as relativistic self-focusing, low-frequency electric and magnetic wake fields generation, the creation of pulse frequency harmonics, nonlinear Compton scattering—while their generating sources are specified. But also new processes are identified, arising from possible couplings among the above quoted effects. Particular emphasis is put on the coupling of self-induced magnetic field with its generating pump, leading itself to the possible magnetic relativistic guiding of light.

Relativistic theory of nuclear forces

A theory of nuclear forces is proposed which results in a nonlinear relativistic equation for their potential. This equation is used to explain nuclear saturation.

Completely integrable model of classical field theory with nontrivial particle interaction in two-dimensional space-time

The complete integrability of a nonlinear relativistic equation, being a generalization of the sine-Gordon equation for the case of group SO(3), is shown. The soliton spectrum is described and exact solutions describing their interaction are obtained. It is shown that together with elastic scattering there exist the processes of decay and merging of solitons. A generalization of the sine-Gordon equation for an arbitrary compact Lie group is discussed.

Self-trapping of intense optical beams in plasmas

Two new results concerning the propagation of intense optical beams in plasmas are presented. The first is an analytic solution of the relativistic nonlinear equations governing the steady-state diffraction of intense, circularly polarized, electromagnetic beams in warm, quasi-neutral plasmas. The beams cannot self-focus, but can self-trap (propagate without diverging) at powers above Pc=2×105 (λD/λ)2 MW, where λD/λ is the Debye length/vacuum optical wavelength ratio. The second result is an exact solution of the relativistic nonlinear equations for a self-trapped beam propagating with constant cross section; this solution yields self-consistent density and intensity profiles and a relativistic beam-plasma dispersion relation.

The motion of a charged particle in electrostatic and electromagnetic fields

Extract solutions are given of the relativistic non-linear equations of motion for both the momentum and displacement of a charged particle which is injected with given momentum into an arbitrary number of intense electromagnetic waves of any polarization and frequency (including zero) which propagate along a uniform electrostatic field with the speed of light.

Estimates for the efficient production of antihydrogen by lasers of very high intensities

Starting from the non-linear relativistic equations of motion for charged particles in the very high intensity fields of laser radiation, the maximum kinetic energy ɛkin of the resulting oscillation is derived exactly. In non-relativistic conditions ɛkin agrees with the well-known valuee2Ev2/(2m0ω2|n|), showing a dependence on the rest massm0 of the particle. In the relativistic case, the mass dependence vanishes. The multipole radiation is calculated on the basis of Sommerfeld's formula for relativistic conditions. It is shown that this radiation is not important for oscillation energies up to\( \in _{kin^{mr} } \)=70m0c2 for electrons in neodymium glass laser radiation and up to higher values for CO2 lasers and for protons. With the limitationm0c2 < ekin <\( \in _{kin^{mr} } \), the formula for ɛkin is used to calculate the pair production (a) for singly oscillating particles in vacuum without collisions and (b) for plasmas with collisions. Taking into account the local increase of the effective electric laser field near the cut-off density due to the decrease of ¦n¦ (n is the complex refractive index), there is the possibility of efficient proton pair production at intensities of 1019 W cm−2 for neodymium glass lasers and of 1017 W cm−2 for CO2 lasers, besides electron pair production.