Equal energy phase space trajectories in resonant wave interactions

Adiabatic evolution of two and three resonantly interacting wave systems with nonlinear frequency/wave vector shifts is discussed. The corresponding Hamiltonian, depending on the coupling, detuning, and nonlinear frequency shift parameters may have a variable number of fixed points, i.e., the system can experience a topological change of phase space when these parameters vary in time or space. It is shown that the oscillation periods of two equal energy trajectories in these wave systems are equal and the difference between the action integrals of such trajectories is obtained analytically as a function of the system parameters. Based on these findings, a scheme of simultaneous adiabatic variation in the parameters is designed, such that any pair of initially equal energy trajectories continues to have the same energy at later times. These results are generalizations of a previous work [O. Polomarov and G. Shvets, Phys. Plasmas 13, 054502 (2006)] for a single, resonantly driven wave.