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Fluid Dynamics Seminar

Monday, Sep 18, 2006, 4:15 PM
Cullimore Lecture Hall, Room 611
New Jersey Institute of Technology

On the two-dimensional (2-D) Benjamin model equation

Boguk Kim

Department of Mathematical Sciences, NJIT

Abstract

The Benjamin equation, derived from the Euler equations, models the dispersive wave motion of weakly-nonlinear long-wave interfacial elevation between two immiscible potential flow layers --- with a thin fluid layer on the top of a heavier deep fluid, when a strong interfacial tension is present. As dispersive terms, the model equation has both the Hilbert transform, which accounts for the gravitational effect in deep water, and the capillary term of the Korteweg-de Vries (KdV) type. The spatially one-dimensional (1-D) model equation is first derived by T. J. Benjamin (JFM 1970), and he showed that it allows one parameter family of plane solitary wave solutions. In this talk, we extend the 1-D model equation by including long-wave dispersive effect in the transverse direction to the dominant wave propagation in the same physical setting considered by T. J. Benjamin, so that we derive a new model equation, named 2-D Benjamin. A newly added transverse dispersive term turns out to be of the same type as the one in the Kadomtsev-Petviashvili I (KP-I) equation, which is a well-known two-dimensional extension to the KdV equation for the long wave dynamics that occurs on the surface of shallow water with the presence of a strong surface tension. We also discover a new kind of solitary wave solutions for the 2-D Benjamin equation, which are fully localized (lump-type) in two dimensions. The solutions feature a pitchfork bifurcation at a nonzero critical wavenumber where the minimum of linear wave speed is attained by an entirely similar fashion to the generation mechanism of the corresponding plane solitary waves: at the bifurcation point, solitary waves form weakly nonlinear wavepackets, whose envelopes are essentially originated from the localized modes of the radial symmetric nonlinear Schrodinger equation (NLS). Unsteady numerical simulations for the 2-D Benjamin model equation are presented to discuss the stability of those solitary waves lumps, the formation of oblique lumps, and their mutual interactions. Finally, we explore the possibility of translating the whole idea presented here into 2-D model equation scenarios for other analogous fluid waves, such as fully-nonlinear internal waves in a two-fluid system (W. Choi & R. Camassa, JFM 1999) and gravity-capillary waves in fluid layers under normal electric fields (D. T. Papageorgiou, P. G. Petropoulos, & J-M Vanden-Broeck, Phys. Rev. E 2005).