Chin. Phys., 2007, Vol. 16(12): 3589-3594    DOI: 10.1088/1009-1963/16/12/006
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# A kind of extended Korteweg--de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

Yang Lian-Guia, Song Jin-Baob, Yang Hong-Lic, Liu Yong-Junc
a Department of Mathematics, Inner Mongolia University, Hohhot 010021, China; b Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China; c Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China;Graduate School, Chinese Academy of Sciences, Beijing 100049, China
Abstract  This paper considers interfacial waves propagating along the interface between a two-dimensional two-fluid with a flat bottom and a rigid upper boundary. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. It just focuses on the weakly non-linear small amplitude waves by introducing two small independent parameters: the nonlinearity ratio $\varepsilon$, represented by the ratio of amplitude to depth, and the dispersion ratio $\mu$, represented by the square of the ratio of depth to wave length, which quantify the relative importance of nonlinearity and dispersion. It derives an extended KdV equation of the interfacial waves using the method adopted by Dullin {\it et al} in the study of the surface waves when considering the order up to $O(\mu ^2)$. As expected, the equation derived from the present work includes, as special cases, those obtained by Dullin {\it et al} for surface waves when the surface tension is neglected. The equation derived using an alternative method here is the same as the equation presented by Choi and Camassa. Also it solves the equation by borrowing the method presented by Marchant used for surface waves, and obtains its asymptotic solitary wave solutions when the weakly nonlinear and weakly dispersive terms are balanced in the extended KdV equation.
Keywords:  two-fluid system      solitary wave solution      interfacial waves      extended KdV equation
Published:  20 December 2007
 PACS: 05.45.Yv (Solitons) 02.30.Jr (Partial differential equations) 47.35.-i (Hydrodynamic waves)
Fund: Project supported by the National Science Fund for Distinguished Young Scholars (Grant No 40425015).