Analytical solutions for nonlinear plasma waves with time-varying complex frequency

Abstract

Bernstein-Kruskal-Greene (or BGK) modes are ubiquitous nonlinear solutions for the 1D electrostatic Vlasov equation, with the particle distribution function f given as a function of the particle energy. Here, we consider other solutions f = f [ ] where the particle energy is equal to the second-order velocity space Taylor expansion of the function (x, v, t) near the wave-particle resonance. This formalism allows us to analytically examine the time evolution of plasma waves with time-varying complex frequency ω(t) + iγ(t) in the linear and nonlinear phases. Using a Laplace-like decomposition of the electric potential, we give allowed solutions for the time-varying complex frequencies. Then, we show that f can be represented analytically via a family of basis decompositions in such a system. Using a Gaussian decomposition, we give approximate solutions for contours of constant f for a single stationary frequency mode, and derive the evolution equation for the nonlinear growth of a frequency sweeping mode. For this family of modes, highly nonlinear orbits are found with the effective width in velocity of the island roughly a factor of √ 2 larger than the width of a BGK island.

Island formation in collisionless kinetic plasmas with perturbed orbits

“…. 22 Here, we show that the first-order nonlinear ODE can be reformulated instead as defining an energy-like quantity , given by the sum of the single particle energy and a generating function ψ. We show that the orbit period τ can be represented as an adiabatic part (which one can identify as the unperturbed orbit), and a perturbation.…”

Nonlinear harmonics coupled by parallel wave propagations in a time-dependent plasma flow

“…However, plasmas involve nonlinearly coupled fields and associated particle motions [17][18][19][20]. Even when the intensity of a nonlinearly excited wave is small compared to a linear wave, the nonlinear wave can carry information of a plasma or trigger instabilities [17,[21][22][23]. By investigating the nonlinearity, one can understand plasma kinetics which cannot be explained by linear theory.…”