The stability of finite amplitude, two-dimensional solitary waves of permanent form in water of uniform depth with respect to two-dimensional infinitesimal disturbances is investigated. It is numerically confirmed that the recent analytical results obtained by Saffman (submitted to J. Fluid Mech.) for the ‘‘superharmonic’’ instability of periodic waves hold also in the case of solitary waves.

The temporal evolution of nonlinear wave fields of surface gravity waves is studied by large-scale direct numerical simulations of primitive equations in order to verify Hasselmann's theory for nonlinear energy transfer among component gravity waves. In the simulations, all the nonlinear interactions, including both resonant and non-resonant ones, are taken into account up to the four-wave processes. The initial wave field is constructed by combining more than two million component free waves in such a way that it has the JONSWAP or the Pierson–Moskowitz spectrum. The nonlinear energy transfer is evaluated from the rate of change of the spectrum, and is compared with Hasselmann's theory. It is shown that, in spite of apparently insufficient duration of the simulations such as just a few tens of characteristic periods, the energy transfer obtained by the present method shows satisfactory agreement with Hasselmann's theory, at least in their qualitative features.

Reflection of an obliquely incident solitary wave by a vertical wall is studied numerically by applying the ‘high-order spectral method’ developed by Dommermuth & Yue (1987). According to the analysis by Miles (1977a, b) which is valid when ai [Lt ] 1, the regular type of reflection gives way to ‘Mach reflection’ when ai/(3ai)½ ≤ 1, Where ai is the amplitude of the incident wave divided by the quiescent water depth d and ψi is the angle of incidence. In Mach reflection, the apex of the incident and the reflected waves moves away from the wall at a constant angle (ψ*, say), and is joined to the wall by a third solitary wave called ‘Mach stem’. Miles model predicts that the amplitude of Mach stem, and so the run-up at the wall, is 4ai when ψi = (3ai)½.Our numerical results shows, however, that the effect of large amplitude tends to prevent the Mach reflection to occur. Even when the Mach reflection occurs, it is ‘contaminated’ by regular reflection in the sense that all the important quantities that characterize the reflection pattern, such as the stem angle ψ*, the angle of reflection ψr, and the amplitude of the reflected wave ar, are all shifted from the values predicted by Miles’ theory toward those corresponding to the regular reflection, i.e. ψ* = 0, ψr = ψi, and ar = ai. According to our calculations for ai = 0.3, the changeover from Mach reflection to regular reflection happens at ψi ≈ 37.8°, which is much smaller than (3ai)½ = 54.4°, and the highest Mach stem is observed for ψi = 35° (ψi/(3ai)½ = 0.644). Although the ‘four-fold amplification’ is not observed for any value of ψi considered here, it is found that the Mach stem can become higher than the highest two-dimensional steady solitary wave for the prescribed water depth. The numerical result is also compared with the analysis by Johnson (1982) for the oblique interaction between one large and one small solitary wave, which shows much better agreement with the numerical result than the Miles’ analysis does when ψi is sufficiently small and the Mach reflection occurs.

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We propose a procedure to grow GaN quantum dots (QDs) on AlN by using the Ga surfactant effect in plasma-assisted molecular beam epitaxy. Self-formed GaN islands were spontaneously generated under vacuum, after evaporation of the Ga bilayer stabilizing the two-dimensional GaN layer grown under Ga-rich conditions. Island characteristics (size and density) are studied as a function of the nominal amount of GaN deposited. We demonstrate that the QD density can be controlled in the 3×1010 cm−2–2×1011 cm−2 range. It is shown that beyond a given amount of GaN nominally deposited, there is a coexistence between elastic and plastic relaxation, with GaN islands being formed on a partially relaxed two-dimensional GaN layer thicker than two monolayers.