Equal energy phase space trajectories in resonant wave interactions

Adiabatic evolution of a nonlinear resonantly driven wave system and two/three coupled waves system generic to a variety of plasma physics problems is studied. The corresponding Hamiltonian, depending on the coupling, detuning and nonlinear frequency shift parameters has a variable number of fixed points. The system can be bistable due to repeated separatrix crossing in the phase space. It is analytically shown that the oscillation periods along trajectories corresponding to the same value of the Hamiltonian are equal, and the difference of the corresponding areas under them is obtained as a function of the system parameters. A scheme of simultaneous adiabatic change of system parameters is constructed, in such a way that any pair of trajectories that have equal energy at some time will continue to have the same energy anytime. These results are generalizations of a previous work [O. Polomarov and G. Shvets, Phys. Plasmas, 13, 054502 (2006)] for a single resonantly driven wave.