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 Schrödinger equations.

The stability of a train of nonlinear surface gravity waves in deep water in the presence of wind blowing over water is considered. An evolution equation is derived for the wave envelope that is correct to fourth order in the wave steepness. The importance of the fourth-order term in the evolution equation was pointed out by Dysthe [Proc. R. Soc. London Ser. A 369, 105 (1979)]. From this evolution equation the expressions for the maximum growth rate of the instability and the frequency at marginal stability are derived, and graphs are plotted for those two expressions against the wave steepness.

A fourth-order nonlinear evolution equation is derived for a wave propagating at the interface of two superposed fluids of infinite depths in the presence of a basic current shear. On the basis of this equation a stability analysis is made for a uniform wave train. Discussions are given for both an air-water interface and a Boussinesq approximation. Significant deviations are noticed from the results obtained from the third-order evolution equation, which is the nonlinear Schrodinger equation. In the Boussinesq approximation, it has been possible to compare the present results with the exact numerical analysis of Pullin and Grimshaw [12], and they are found to agree rather favourably.

We derive a fourth-order nonlinear evolution equation (NLEE) for narrow-banded Stokes wave in finite depth in the presence of surface tension and a mean flow with constant vorticity. The two-dimensional capillary-gravity wave motion on the surface of finite depth is considered here. The analysis is limited to one horizontal dimension, parallel to the direction of wave propagation, in order to take advantage of a formulation using potential flow theory. This evolution equation is then employed to examine the effect of vorticity on the Benjamin-Feir instability (BFI) of the Stokes capillary-gravity wave trains. It is found that the vorticity modifies significantly the modulational instability and in the case of finite depth the combined effect of vorticity and capillarity is to enhance the instability growth rate influenced by capillarity when the vorticity is negative. The key point is that the present fourth-order analysis exhibits considerable deviations in the stability properties from the third-order analysis and gives better results consistent with the exact numerical results. Furthermore, the influence of linear shear current on Peregrine breather (PB) is studied.

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