A kind of extended Korteweg--de Vries equation  and solitary wave solutions for interfacial waves in a two-fluid system

doi:10.1088/1009-1963/16/12/006

中国物理B ›› 2007, Vol. 16 ›› Issue (12): 3589-3594.doi: 10.1088/1009-1963/16/12/006

A kind of extended Korteweg--de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

杨联贵¹, 宋金宝², 杨红丽³, 刘永军³

(1)Department of Mathematics, Inner Mongolia University, Hohhot 010021, China; (2)Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China; (3)Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China;Graduate School, Chinese Academy of Sciences, Beijing 100049, China

出版日期:2007-12-20 发布日期:2007-12-20
基金资助:
Project supported by the National Science Fund for Distinguished Young Scholars (Grant No 40425015).

A kind of extended Korteweg--de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

Yang Hong-Li(杨红丽)^a)c)^†, Song Jin-Bao(宋金宝)^a), Yang Lian-Gui(杨联贵)^b), and Liu Yong-Jun(刘永军)^a)c)

^a Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China; ^b Department of Mathematics, Inner Mongolia University, Hohhot 010021, China; ^cGraduate School, Chinese Academy of Sciences, Beijing 100049, China

Online:2007-12-20 Published:2007-12-20
Supported by:
Project supported by the National Science Fund for Distinguished Young Scholars (Grant No 40425015).

摘要/Abstract

摘要： 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.

关键词: two-fluid system, interfacial waves, extended KdV equation, solitary wave solution

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 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 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.

Key words: two-fluid system, interfacial waves, extended KdV equation, solitary wave solution

中图分类号: (Solitons)

05.45.Yv

02.30.Jr (Partial differential equations) 47.35.-i (Hydrodynamic waves)

杨红丽, 宋金宝, 杨联贵, 刘永军. A kind of extended Korteweg--de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system[J]. 中国物理B, 2007, 16(12): 3589-3594.

Yang Hong-Li(杨红丽), Song Jin-Bao(宋金宝), Yang Lian-Gui(杨联贵), and Liu Yong-Jun(刘永军). A kind of extended Korteweg--de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system[J]. Chinese Physics, 2007, 16(12): 3589-3594.

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A kind of extended Korteweg--de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

A kind of extended Korteweg--de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

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