Nonlinear Dispersion of Stationary Waves in Collisionless Plasmas

Abstract

A nonlinear dispersion of a general stationary wave in collisionless plasma is obtained in a nondifferential form expressed in terms of a single-particle oscillation-center Hamiltonian. For electrostatic oscillations in nonmagnetized plasma, considered as a paradigmatic example, the linear dielectric function is generalized, and the trapped particle contribution to the wave frequency shift Δω is found analytically as a function of the wave amplitude a. Smooth distributions yield Δω ∼ a(1/2), as usual. However, beamlike distributions of trapped electrons result in different power laws, or even a logarithmic nonlinearity, which are derived as asymptotic limits of the same dispersion relation.

Axiomatic geometrical optics, Abraham-Minkowski controversy, and photon properties derived classically

“…Note also that similar calculations can be performed for nonlinear waves too, for which L can be constructed from first principles as well [106][107][108][109].…”

“…-Consider an isotropic medium (such as gas, fluid, or plasma) comprised of elementary [139] particles or fluid elements whose dynamics absent a wave is described by some aggregate Lagrangian L. In the presence of a wave, the WMS Lagrangian is hence L + L, where L = L dV is the wave Lagrangian. Assuming that particles contribute to L additively, the latter can be written as L = L (0) − ℓ Φ (ℓ) , where L (0) is independent of all particle velocities u (ℓ) , and each of the so-called ponderomotive potentials Φ (ℓ) [106,107], or dipole potentials [67,68], depends on the specific u (ℓ) but not on other velocities. Omitting the index ℓ, we can write the canonical momentum of each particle as the sum of the mechanical part ∂ u L and the ponderomotive part −∂ u Φ, also yielding the ponderomotive contribution to the particle canonical energy, −u · ∂ u Φ.…”

Axiomatic Geometrical Optics, Abraham-Minkowski Controversy, and Photon Properties Derived Classically

“…-Consider an isotropic medium (such as gas, fluid, or plasma) comprised of elementary [136] particles or fluid elements whose dynamics absent a wave is described by some aggregate Lagrangian L. In the presence of a wave, the WMS Lagrangian is hence L + L, where L = L dV is the wave Lagrangian. Assuming that particles contribute to L additively, the latter can be written as L = L (0) − Φ ( ) , where L (0) is independent of all particle velocities u ( ) , and each of the so-called ponderomotive potentials Φ ( ) [104,105], or dipole potentials [65,66], depends on the specific u ( ) but not on other velocities. Omitting the index , we can write the canonical momentum of each particle as the sum of the mechanical part ∂ u L and the ponderomotive part −∂ u Φ, also yielding the ponderomotive contribution to the particle canonical energy, −u · ∂ u Φ.…”

“…is that in vacuum [104],Ẽ andB are the electric and magnetic field envelopes, and U is the potential energy density of the wave-medium interaction [cf. Eq.…”

“…[104,105], this implies assigning the following ponderomotive potentials to particles (or fluid elements) comprising the medium:…”