Interfacial waves with free-surface boundary conditions: An approach via amodel equation

In a two-fluid system where the lower fluid is bounded below by a rigid bot tom and the upper fluid is bounded above by a free surface, two kinds of so litary waves can propagate along the interface and the free surface: classi cal solitary waves characterized by a solitary pulse or generalized solitar y waves with nondecaying oscillations in their tails in addition to the sol itary pulse. The classical solitary waves move faster than the generalized solitary waves. The origin of the nonlocal solitary waves can be understood from a physical point of view. The dispersion relation for the above syste m shows that short waves can propagate at the same speed as a "slow" solita ry wave. The interaction between the solitary wave and the short waves crea tes a nonlocal solitary wave. In this paper, the interfacial-wave problem i s reduced to a system of ordinary differential equations by using a classic al perturbation method, which takes into consideration the possible resonan ce between short waves and "slow" solitary waves. In the past, classical Ko rteweg-de Vries type models have been derived but cannot deal with the reso nance. All solutions of the new system of model equations, including classi cal as well as generalized solitary waves, are constructed. The domain of v alidity of the model is discussed as well. It is also shown that fronts con necting two conjugate states cannot occur for "fast" waves. For "slow" wave s, fronts exist but they have ripples in their tails. (C) 2001 Elsevier Sci ence B.V. All rights reserved.