Projection-operator method for the nonlinear three-wave interaction

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Abstract
A theory of nonlinear three-wave interaction is presented, where a finite bandwidth of the interacting waves is considered (Δω≠0). Dissipative processes are neglected, and a Hamiltonian formulation is used. The evolution equations for the wave intensities are obtained with the aid of a projection-operator method, similar to those used in nonequilibrium statistical mechanics, but our formulation is deterministic and no statistical hypothesis is needed. These equations generalize the well-known fixed-phase equations (Δω=0) and are formally analogous to them, if the ballistic and memory effects are neglected.