Stability Analysis from Fourth-Order Nonlinear Multiphase Deep Water Wavetrains

The nonlinear evolution equations (NLEE) of fourth-order for the propagation of two surface gravity wavetrains (SGW) in infinite depth of water are derived. Based on these evolution equations, the properties of sideband instability of uniform gravity waves are investigated. Considerable deviations are observed from the results obtained from the cubic NLEE, which are the nonlinear Schrödinger equations (NLSE). An expression of the instability growth rate is obtained, and this expression shows a key result that the fourth-order terms in the evolution equations significantly modify the modulational instability properties and produce a decrease in the growth rate for the case of acute angle.