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International Journal of Non-Linear Mechanics, Volume 69, 2015, Pages 66-70.
Erinc Ozden, Hilmi Demiray.
Isik University, Department of Mathematics, 34980 Sile, Istanbul, Turkey.
Abstract
Upon discovering the wrongness of the statement “although this term does not cause any secularity for this order it will cause secularity at higher order expansion, therefore, that term must vanish” by Su and Mirie[4], in the present work, we studied the head-on collision of two solitary waves propagating in shallow water by introducing a set of stretched coordinates in which the trajectory functions are of order of ϵ2, where ϵ is the smallness parameter measuring non-linearity. Expanding the field variables and trajectory functions into power series in ϵ, we obtained a set of differential equations governing various terms in the perturbation expansion. By solving them under non-secularity condition we obtained the evolution equations and also the expressions for phase functions. By seeking a progressive wave solution to these evolution equations we have determined the speed correction terms and the phase shifts. As opposed to the result of Su and Mirie [4] and similar works, our calculations show that the phase shifts depend on both amplitudes of the colliding waves.
Significance Statement
Realizing the incorrect statement about removing the secularity in the solution of head-on collision problem of two solitary waves propagating in shallow water by Su and Mirie( reference [4] in the main text), we re-examined the head-on collision of two solitary waves in shallow water by introducing a set of stretched coordinates, in which the trajectory functions are of order of ϵ2, where ϵ is the smallness parameter measuring the nonlinearity and the dispersion. Expanding the field variables and the trajectory functions into power series in ϵ, a set of differential equations involving various terms in the perturbation expansion are obtained. By solving these differential equations under non-secularity conditions, various evolution equations and some expressions for the trajectory functions are obtained. Seeking a progressive wave solution to these evolution equations we have determined the speed correction terms and the phase shifts.
*As opposed to the result of Sue and Mirie, our calculations reveal that the trajectory functions and the phase shifts of the colliding waves are of order of ϵ2, rather than ϵ.
* It is further observed that the phase shifts depend on the amplitudes of both colliding waves.
