Theoretical models describing the slow space and time evolution of weakly nonlinear surface gravity waves are generally based on the two assumptions: that the steepness is small k0a 1, and that the bandwidth is narrow |Δk|/k0 1. Typically, one assumes that the steepness and the bandwidth are of the same order of magnitude such that the leading nonlinear and dispersive effects balance at the third order The resulting amplitude modulation equation is known as the nonlinear Schrödinger equation (NLS). This equation was pioneered by Benney & Newell (1967) for nonlinear dispersive waves in general, by Zakharov (1968), Hasimoto & Ono (1972) and Davey (1972) for gravity waves on deep water, and by Benney & Roskes (1969) for gravity waves on finite depth. A modification of the nonlinear Schrödinger equation to fourth-order accuracy which we shall denote the MNLS equation, was derived by Dysthe (1979) for gravity waves on infinite depth, with minor modifications for gravity waves on deep water by Lo & Mei (1985), and by Brinch-Nielsen & Jonsson (1986) for gravity waves on finite depth. We here define finite depth, deep water and infinite depth as (k0h)–1 being and 0, respectively.
The MNLS equation has successfully been used to model several aspects of the long-time evolution of weakly nonlinear narrow-banded water waves. It predicts the asymmetric growth of upper and lower sidebands, accompanied by the forward steepening of initially symmetric wave-groups as reported in the experiment of Feir (1967). Enhanced with a highly simplified model for wave damping due to breaking, it was shown by Trulsen & Dysthe (1990) that the MNLS equation can predict the permanent downshift of the carrier wave frequency, which was observed experimentally by Lake et al. (1977). Furthermore, with the additional enhancement of a model for wind growth due to Plant (1982) and damping by wave breaking, it was shown by Trulsen & Dysthe (1992) that the MNLS equation can also predict that a strong wind can stabilize Stokes waves such that modulational instability and frequency downshift are suppressed. This was observed experimentally by Bliven et al. (1986). The MNLS equation was employed by Hara & Mei (1991) to predict downshift of the carrier wave frequency for waves forced by a weak wind and damped by eddy viscosity. There has also been recent work further investigating the downshift by the addition of damping terms to the MNLS equation (e.g. Uchiyama & Kawahara 1994; Kato & Oikawa 1995). These applications all consider evolution in one horizontal dimension.
For application of the MNLS equation to describe the situation in figures 1 and 2, we observe that the central wave steepness is within the domain of validity of the MNLS equation, but the bandwidth is not. Therefore, an improved model is called for that can describe broader bandwidth wave trains. To this end the Zakharov integral equation (Zakharov 1968; Crawford, Saffman & Yuen 1980; Stiassnie & Shemer 1984) has been developed to avoid the limitation in bandwidth altogether. This additional generality has the price of making the Zakharov equation unnecessarily expensive to solve numerically for the present problem. In order to maintain the relative simplicity of the MNLS equation, it is desirable to look for ways to relax the bandwidth constraint, while keeping the same accuracy in nonlinearity.
Even though the waves in figure 1 are not on deep water, we limit the present discussion to waves on deep water. In section 2 we first review the NLS and MNLS equations, and then summarize a new modified nonlinear Schrödinger equation for broader bandwidths by requiring while keeping the same accuracy in nonlinearity (Trulsen & Dysthe 1996). The new equation will be denoted the BMNLS equation. The resolution in bandwidth can be assessed by comparison between predicted and exact stability results for Stokes waves, and the new equation has been found to be in good conformance with the resolution required by the above order-of-magnitude analysis. An extension to finite depth has been derived by us, and will be reported in the future.
The BMNLS equation can be solved numerically with periodic boundary conditions in two hor