Resonant-test-field model of fluctuating nonlinear waves

A Hamiltonian system of nonlinear dispersive waves is used as a basis for generalizing the test-wave model to a set of resonantly interacting waves. The resonant test field (RTF) is shown to obey a nonlinear generalized Langevin equation in general. In the Markov limit a Fokker-Planck equation is obtained and the exact steady-state solution is determined. An algebraic expression for the power spectral density is obtained in terms of the number of resonantly interacting waves ($n$) in the RTF, the interaction strength ($V_k$), and the dimensionality of the wave field ($d$). For gravity waves on the ocean surface a $k^{-4\mathrm{}}$ spectrum is obtained, and for capillary waves a $k^{-8\mathrm{}}$ spectrum, both of which are in essential agreement with data.