Head-on collision between two mKdV solitary waves in a two-layer fluid system

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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 mKdV equation, 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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    Dai Shi-qiang. Solitary waves at the interface of a two-layer fluid,Appl. Math. and Mech.,3, 6 (1982), 777–788.

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    Dai Shi-qiang, Head-on collision between two interfacial solitary waves,Acta Mechanica Sinica,15 (1983), 623–632.

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    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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Communicated by Dai Shi-quiang

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Yong, Z. Head-on collision between two mKdV solitary waves in a two-layer fluid system. Appl Math Mech 13, 407–417 (1992).

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Key words

  • mKdV solitary wave
  • head-on collision
  • perturbation method