一个两流体系统中mKdV孤立波的迎撞*
Head-on Collision between Two mKdV Solitary Waves in a Two-Layer Fluid System
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摘要: 本文从文[2]的基本方程出发,采用约化摄动方法和PLK方法,讨论了三阶非线性和色散效应相平衡的修正的KdV(mKdV)孤立波迎撞问题.这些波在流体密度比等于流体深度比平方的两流体系统界面上传播.我们求得了二阶摄动解,发现在不考虑非均匀相移的情况下,碰撞后孤立波保持原有的形状,这与Fornberg和whitham[6]的追撞数值分析结果一致,但当考虑波的非均匀相移后,碰撞后波形将变化.Abstract: In this paper, based on the equations presented in [2], the head-on collision between two solitary waves described by the modified KdV equation (the mKdVequation, for short) is investigated by using the reductive perturbation method combined with the PLK method.These waves propagate at the interface of a two-fluid system, in which the density ratio of the two fluids equals the square of the depth ratio of the fluids.The second order perturbation solution is obtained.It is found that in the case of disregarding the nonuniform phase shift, the solitary waves preserve their original profiles after collision, which agrees with Fornberg and Whitham's numerical result of overtaking collision[6] whereas after considering the nonuniform phase shift, the wave profiles may deform after collision.
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Key words:
- mKdV solitary wave /
- head-on collision /
- perturbation method
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[1] Mile,J.W.,Solitary waves,Ann.Rev.Fluid Mech.,12 (1980),11-43. [2] 戴世强.两层流体界面上的孤立波,应用数学和力学,3(6)(1982),721-731 [3] Gear,J.and R.Grimshaw,Weak and strong interaction between internal solitary waves,Studies in Appl.Math.,68 (1984),253-258. [4] Mirie,R.M.and C.H.Su,Internal solitary waves and their head-on collision,Part I,J.Fluid Mech.,147 (1984),213-231;part Ⅱ,Phys.Fluids,29 (1986),31-37. [5] 戴世强,两个界而孤立波之间的迎撞,力学学报,6(1983),523-532 [6] Fornberg,B.and G.B.Whitham,A numerical and theoretical study of certain nonlinear wave phenomena,Phil.Trans.R.Soc.London,289(1978),373-404． -
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