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

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 $f = f[\epsilon]$ where the particle energy is equal to the second-order velocity space Taylor expansion of the function $\epsilon(x,v,t)$ near the wave-particle resonance. This formalism allows us to analytically examine the time evolution of plasma waves with time-varying complex frequency $\omega(t) + i \gamma(t)$ 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 $\sqrt{2}$ larger than the width of a BGK island.