Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water

TLDR: In this paper, the authors derived the fourth-order nonlinear evolution equations for a deepwater surface gravity wave packet in the presence of a second wave packet, where it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap.

Abstract: Fourth‐order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves as first pointed out by Dysthe [Proc. R. Soc. London Ser. A 369, 105 (1979)] and later elaborated by Janssen [J. Fluid Mech. 126, 1 (1983)], are derived for a deep‐water surface gravity wave packet in the presence of a second wave packet. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. Stability analysis is made for a uniform Stokes wave train in the presence of a second wave train. Graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness. Significant deviations are noticed from the results obtained from the third‐order evolution equations which consist of two coupled nonlinear Schrodinger equations.