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Yayın Re-visiting the head-on collision problem between two solitary waves in shallow water(Pergamon-Elsevier Science Ltd, 2015-03) Özden, Ali Erinç; Demiray, HilmiUpon 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 epsilon(2), where epsilon is the smallness parameter measuring non-linearity. Expanding the field variables and trajectory functions into power series in epsilon, 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 Mine [4] and similar works, our calculations show that the phase shifts depend on both amplitudes of the colliding waves.Yayın On head-on collision between two solitary waves in shallow water: the use of the extended PLK method(Springer, 2015-10) Özden, Ali Erinç; Demiray, HilmiIn the present work, we examined the head-on collision of solitary waves in shallow water theory, through the use of extended Poincare–Lighthill–Kuo (PLK) method based on the combination of reductive perturbation method with strained coordinates. Motivated with the result obtained by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), we introduced a set of stretched coordinates that include some unknown functions which are to be determined so as to remove secularities that might occur in the solution. By expanding these unknown functions and the field variables into power series in the smallness parameter ?, introducing them into the field equations and imposing the conditions to remove the secularities, we obtained some evolution equations. By seeking a progressive wave solution to these evolution equations, we determined the speed correction terms and the phase-shift functions. The result obtained here is exactly the same with found by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), wherein the analysis employed by Su and Mirie (J Fluid Mech 98:509–525, 1980) is utilized.Yayın Linearization of second-order jump-diffusion equations(Springer Berlin Heidelberg, 2013-03-01) Özden, Ali Erinç; Ünal, GazanferWe give the exact linearization criterion for the second-order jump-diffusion equations. We also present several illustrative examples.
