Fully nonlinear model equations, including dispersive effects at one-order higher approximation than in the model of Choi & Camassa (J. Fluid Mech., vol. 396, 1999, pp. 1–36), are derived for long internal waves propagating in two spatial horizontal dimensions in a two-fluid system, where the lower layer is of infinite depth. The model equations consist of two coupled equations for the displacement of the interface and the horizontal velocity of the upper layer at an arbitrary elevation, and they are correct to O(μ2) terms, where μ is the ratio of thickness of the upper-layer fluid to a typical wavelength. For solitary waves propagating in one horizontal direction, the two coupled equations reduce to a single equation for the elevation of the interface. Solitary wave profiles obtained numerically from this equation for different wave speeds are in good agreement with computational results based on Euler's equations. A numerical approach for the propagation of solitary waves is provided in the weakly nonlinear case.

From Zakharov's integral equation a nonlinear evolution equation for broader bandwidth gravity waves in deep water is obtained, which is one order higher than the corresponding equation derived by Trulsen and Dysthe ͓"A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water," Wave Motion 24, 281 ͑1996͔͒. The instability regions in the perturbed wave-number space for a uniform Stokes wave obtained from this equation are in surprisingly good agreement with the exact results obtained by McLean et al. ͓"Three dimensional instability of finite amplitude water waves," Phys. Rev. Lett. 46, 817 ͑1981͔͒.

Starting from the Zakharov integral equation, two coupled fourth-order nonlinear equations have been derived for the evolution of the amplitudes of two capillary-gravity wave packets propagating in the same direction. The two evolution equations are used to investigate the stability of a uniform capillary-gravity wave train in the presence of another having the same group velocity. The relative changes in phase velocity of each uniform wave train due to the presence of the other one have been shown in figures for different wave numbers. The condition of instability of a wave of greater wavelength in the presence of a wave of shorter wavelength is obtained. It is observed that the instability region for a surface gravity wave train in the presence of a capillary-gravity wave train expands with the increase of wave steepness of the capillary-gravity wave train. It is found that the presence of a uniform capillary-gravity wave train causes an increase in the growth rate of instability of a uniform surface gravity wave train.

In search for a synthetic understanding, a scenario for the evolution of the star formation rate and the chemical abundances in galaxies is proposed, combining gas infall from galactic halos, outflow of gas by supernova explosions, and an oscillatory star formation process. The oscillatory star formation model is a consequence of the modelling of the fractional masses changes of the hot, warm and cold components of the interstellar medium. The observed periods of oscillation vary in the range (0.1 − 3.0) × 10 7 yr depending on various parameters existing from giant to dwarf galaxies. The evolution of metallicity varies in giant and dwarf galaxies and depends on the outflow process. Observed abundances in dwarf galaxies can be reproduced under fast outflow together with slow evaporation of cold gases into hot gas whereas slow outflow and fast evaporation is preferred for giant galaxies. The variation of metallicities in dwarf galaxies supports the fact that low rate of SNII production in dwarf galaxies is responsible for variation in metallicity in dwarf galaxies of similar masses as suggested by various authors.

For a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.