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Advances in Mathematical Physics
Volume 2017, Article ID 9312681, 8 pages
Research Article

Model Equations for Three-Dimensional Nonlinear Water Waves under Tangential Electric Field

School of Architecture Engineering, Neijiang Normal University, Sichuan 641100, China

Correspondence should be addressed to Bo Tao; moc.qq@345174572

Received 18 May 2017; Revised 18 September 2017; Accepted 11 October 2017; Published 12 November 2017

Academic Editor: Prabir Daripa

Copyright © 2017 Bo Tao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We are concerned with gravity-capillary waves propagating on the surface of a three-dimensional electrified liquid sheet under a uniform electric field parallel to the undisturbed free surface. For simplicity, we make an assumption that the permittivity of the fluid is much larger than that of the upper-layer gas; hence, this two-layer problem is reduced to be a one-layer problem. In this paper, we propose model equations in the shallow-water regime based on the analysis of the Dirichlet-Neumann operator. The modified Benney-Luke equation and Kadomtsev-Petviashvili equation will be derived, and the truly three-dimensional fully localized traveling waves, which are known as “lumps” in the literature, are numerically computed in the Benney-Luke equation.