Numerical study of generalized interfacial solitary waves

Abstract

In a two-fluid system where the lower fluid is bounded below by a rigid bottom and the upper fluid is bounded above by a free surface, two kinds of solitary waves can propagate along the interface and the free surface, classical solitary waves characterized by a solitary pulse or generalized solitary waves with in addition nondecaying oscillations in their tails. In this paper, we present numerical solutions of generalized solitary waves. Since generalized solitary waves cannot be obtained as the continuous limit of long waves, we in fact compute generalized long waves. The effects of capillarity are neglected. The solutions depend on four dimensionless parameters, the layer thickness ratio, the density ratio, the Froude number, and the dimensionless amplitude of the oscillations in the far field. If the amplitude of the oscillations is varied while the other three parameters are kept fixed, two limiting cases are conjectured. As the amplitude of the oscillations decreases towards zero, it reaches a minimum nonzero value, which is exponentially small. On the other hand, as the amplitude of the oscillations is increased, the generalized solitary wave eventually becomes a periodic wave. In other words, the oscillations in the far field grow as large as the solitary pulse.