Stochastic Benjamin-Ono equation and its application to the dynamics of nonlinear random waves.

Abstract

The stochastic Benjamin-Ono equation is introduced, which models the propagation of nonlinear random waves in a two-layer fluid system with and without uneven bottom topography. In the case of the flat bottom, the effect of the external random flow field on the evolution of both soliton and periodic wave is investigated. In particular, the mean value and the correlation function of these nonlinear wave fields are calculated exactly under the assumption that the flow obeys the Gaussian stochastic process with a white noise. It is found that in the limit of large time, the mean value of an algebraic soliton approaches a Gaussian wave packet whereas that of a periodic wave is represented by Jacobi's theta function. In the case of the uneven bottom, a perturbation analysis is performed to evaluate the mean value of an algebraic soliton under the influence of random change of bottom topography. The large time asymptotic of the soliton is shown to exhibit a Gaussian wave packet with a small amount of the phase shift caused by the interaction between the soliton and the random bottom topography.