  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.
