We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger ͑NLS͒ equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance ͑the ''figure 8'' of Phillips͒ even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.

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Unstable thin liquid films on solid substrates dewet by hole nucleation on defects or by a linear surface instability (spinodal dewetting). A system with destabilizing short-range and stabilizing long-range molecular interactions is investigated. We show that, for a subrange within the linearly unstable film thickness range, nucleation determines the final structure, whereas spinodal dewetting is of negligible influence. The results are also applicable to the spinodal decomposition of binary mixtures.

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We consider a thin layer of a viscous fluid flowing down a uniformly heated planar wall. The heating generates a temperature distribution on the free surface which in turn induces surface tension gradients. We model this thermocapillary flow by using the Shkadov integral-boundary-layer (IBL) approximation of the Navier–Stokes/energy equations and associated free-surface boundary conditions. Our linear stability analysis of the flat-film solution is in good agreement with the Goussis & Kelly (1991) stability results from the Orr–Sommerfeld eigenvalue problem of the full Navier–Stokes/energy equations. We numerically construct nonlinear solutions of the solitary wave type for the IBL approximation and the Benney-type equation developed by Joo et al. (1991) using the usual long-wave approximation. The two approaches give similar solitary wave solutions up to an $O(1)$ Reynolds number above which the solitary wave solution branch obtained by the Joo et al. equation is unrealistic, with branch multiplicity and limit points. The IBL approximation on the other hand has no limit points and predicts the existence of solitary waves for all Reynolds numbers. Finally, in the region of small film thicknesses where the Marangoni forces dominate inertia forces, our IBL system reduces to a single equation for the film thickness that contains only one parameter. When this parameter tends to zero, both the solitary wave speed and the maximum amplitude tend to infinity.

The process of dewetting of a thin liquid film is usually described using a long-wave approximation yielding a single evolution equation for the film thickness. This equation incorporates an additional pressure term-the disjoining pressure-accounting for the molecular forces. Recently a disjoining pressure was derived coupling hydrodynamics to the diffuse interface model [L. M. Pismen and Y. Pomeau, Phys. Rev. E 62, 2480 (2000)]. Using the resulting evolution equation as a generic example for the evolution of unstable thin films, we examine the thickness ranges for linear instability and metastability for flat films, the families of stationary periodic and localized solutions, and their linear stability. The results are compared to simulations of the nonlinear time evolution. From this we conclude that, within the linearly unstable thickness range, there exists a well defined subrange where finite perturbations are crucial for the time evolution and the resulting structures. In the remainder of the linearly unstable thickness range the resulting structures are controlled by the fastest flat film mode assumed up to now for the entire linearly unstable thickness range. Finally, the implications for other forms of disjoining pressure in dewetting and for spinodal decomposition are discussed.

Conditions for thermodynamic equilibrium of liquid drops on solid substrates are presented. It is shown that if surface force (disjoining/conjoining Derjaguin pressure) action in a vicinity of the three-phase contact line is taken into account the condition of thermodynamic equilibrium is duly satisfied. Then the thermodynamic expressions for equilibrium contact angles of drops on solid substrates and menisci in thin capillaries are expressed in terms of the corresponding Derjaguin isotherm. It is shown that equilibrium contact angles of drops vary significantly depending on the vapour pressure in the ambient atmosphere, while there is a single, unique equilibrium contact angle in thin capillaries. It is also shown that the static advancing contact angle of a drop depends on its volume, in agreement with experimental data. In the case of menisci in capillaries, the expression for the receding contact angle is deduced, with results that are also in agreement with known experimental data.