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.
Fund: Project
supported by the National Science Fund for Distinguished Young
Scholars (Grant No 40425015).
Cite this article:
Yang Lian-Gui, Song Jin-Bao, Yang Hong-Li, Liu Yong-Jun A kind of extended Korteweg--de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system 2007 Chin. Phys. 16 3589
