Analytical evolution of coupled, driven waves near their instability threshold

The nonlinear collisional dynamics of two coupled driven waves in the presence of dissipation is studied analytically within kinetic theory. Sufficiently near marginal stability, time delays become unimportant and the system dynamics are shown to be governed by two first-order coupled autonomous differential equations of cubic order for the wave amplitudes and two complementary first-order equations for the evolution of the phases. It is found that the system of equations can be decoupled and further simplified to a single second-order differential equation of Liénard's type for each amplitude. Numerical solutions for this equation are obtained in the general case while analytic solutions are obtained for special cases in terms of parameters related to the spacing of the resonances of the two waves in frequency space, e.g., wave lengths and the oscillation frequencies. These parameters are further analyzed to find limiting cases of pulsating and quasi-steady saturation. Similarly, to classify equilibrium points, local stability analysis is applied, and bifurcation conditions are determined. The results can be generalized to an n-wave system, where sufficiently near marginal stability, the evolution of the underlying distribution function is shown to naturally reduce to a quasilinear transport regime.