The nonlinear three-wave interaction with a finite spectral width

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Abstract
The nonlinear interaction of three waves propagating in an infinite plasma with a finite spectral bandwidth is studied. A Hamiltonian formulation of the interaction is used and, with the help of a projection operator method, generalized Langevin equations are derived for the wave field amplitudes. A simple form of the evolution equations, more complete than the usual fixed‐phase equations, is obtained when the ballistic term and higher‐order corrections of the memory effects in the Langevin equations are neglected. Approximate analytical solutions are derived using a multiple time scale method. These are compared with the results of numerical integration. A number of new qualitative features related to the finite spectral bandwidth are discussed in detail.