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Yayın A note on the amplitude modulation of symmetric regularized long-wave equation with quartic nonlinearity(Springer, 2012-12) Demiray, HilmiWe study the amplitude modulation of a symmetric regularized long-wave equation with quartic nonlinearity through the use of the reductive perturbation method by introducing a new set of slow variables. The nonlinear Schrodinger (NLS) equation with seventh order nonlinearity is obtained as the evolution equation for the lowest order term in the perturbation expansion. It is also shown that the NLS equation with seventh order nonlinearity assumes an envelope type of solitary wave solution.Yayın A note on the exact travelling wave solution to the KdV-Burgers equation(Elsevier Science, 2003-10) Demiray, HilmiIn the present note, by use of the hyperbolic tangent method, a progressive wave solution to the Korteweg-de Vries-Burgers (KdVB) equation is presented. The solution we introduced here is less restrictive and comprises some solutions existing in the current literature (see [Wave Motion 11 (1989) 559; Wave Motion 14 (1991) 369]).Yayın A study of higher order terms in shallow water waves via modified PLK method(Walter De Gruyter GMBH, 2014-04) Demiray, HilmiIn this work, by utilizing the modified PLK (Poincare-Lighthill-Kou) method, we studied the propagation of weakly nonlinear waves in a shallow water theory and obtained the Korteweg-deVries (KdV) and the linearized KdV equations with non-homogeneous term as the governing equations of various order terms in the perturbation expansion. The result obtained here is exactly the same with that of Kodama and Taniuti [6], who employed the so-called "re-normalization method". Seeking a progressive wave solution to these evolution equations we obtained the speed correction terms so as to remove some possible secularities. The result obtained here is consistent with the results of Demiray [12], wherein the modified reductive perturbation method had been utilized.Yayın An application of the modified reductive perturbation method to a generalized boussinesq equation(Walter De Gruyter GMBH, 2013-02) Demiray, HilmiIn this work, we apply "the modified reductive perturbation method" to the generalized Boussinesq equation and obtain various form of generalized KdV equations as the evolution equations. Seeking a localized travelling wave solutions for these evolution equations we determine the scale parameters g(1) and g(2), which corresponds to the correction terms in the wave speed, so as to remove the possible secularities that might occur. Depending on the sign and the values of certain parameters the resulting solutions are shown to be a solitary wave or a periodic solution. The suitability of the method is also shown by comparing the results with the exact travelling wave solution for the generalized Boussinesq equation.Yayın Modulation of generalized symmetric regularized long-wave equation: generalized nonlinear Schrödinger equation(Freund Publishing House Ltd, 2010-12) Demiray, HilmiIn this work, the application of "the modified reductive perturbation method" is extended to the generalized symmetric regularized long-wave equation for strongly dispersive case and the contribution of higher order terms in the perturbation expansion is obtained. It is shown that the first order term in the perturbation expansion is governed by the generalized nonlinear Schrödinger quation whereas the second order term is governed by the generalized linear Schrödinger equation with a nonhomogeneous term. A travelling wave type of solution to these evolution equations is also given.Yayın A modified reductive perturbation method as applied to nonlinear ion-acoustic waves(Physical Society Japan, 1999-06) Demiray, HilmiThe basic equations describing the nonlinear ion-acoustic waves in a cold collisionless plasma, in the longwave limit, is re-examined through the use of a modified reductive perturbation method. Introducing the concept of a scale parameter and expanding the variables and the scale parameter into a power series of the smallness parameter epsilon, a set of evolution equations is obtained for various order terms in the perturbation expansion. To illustrate the present derivation, a localized travelling wave solution is studied for the derived field equations and the result is compared with those of obtained by Sugimoto and Kakutani(3)) who introduced some slow scales, Kodama and Taniuti(4)) who employed the renormalization method and Malfliet and Wieers,(6)) who employed the dressed solitary wave approach from the outset of their study.Yayın A complex travelling wave solution to the KdV-Burgers equation(Elsevier Science bv, 2005-09-19) Demiray, HilmiIn the present work, making use of the hyperbolic tangent method a complex travelling wave solution to the KdV-Burgers equation is obtained. It is observed that the real part of the solution is the combination of a shock and a solitary wave whereas the imaginary part is the product of a shock with a solitary wave. By imposing some restrictions on the field variable at infinity, two complex waves, i.e., right going and left going waves with specific wave speed are obtained.Yayın Nonlinear wave modulation in a fluid-filled elastic tube with stenosis(Verlag Z Naturforsch, 2008-01) Demiray, HilmiIn the present work, treating the arteries as a thin-walled and prestressed elastic tube with a stenosis and the blood as a Newtonian fluid with negligible viscosity, we have studied the amplitude modulation of nonlinear waves in such a composite system by use of the reductive perturbation method. The a governing evolution equation was obtained as the variable coefficient nonlinear Schrodinger (NLS) equation. By setting the stenosis function equal to zero, we observed that this variable coefficient NLS equation reduces to the conventional NLS equation. After introducing a new dependent variable and a set of new independent coordinates, we reduced the evolution equation to the conventional NLS equation. By seeking a progressive wave type of solution to this evolution equation we observed, that the wave trajectories are not straight lines anymore; they are rather some curves in the (xi, tau) plane. It was further observed that the wave speeds for both enveloping and harmonic waves are variable, and the speed of the enveloping wave increases with increasing axial distance, whereas the speed of the harmonic wave decreases with increasing axial coordinates. The numerical calculations indicated that the speed of the harmonic wave decreases with increasing time parameter, but the sensitivity of wave speed to this parameter is quite weak.Yayın Contribution of higher order terms in nonlinear ion-acoustic waves: strongly dispersive case(Physical Soc Japan, 2002-08) Demiray, HilmiContribution of higher order terms in the perturbation expansion for the strongly dispersive ion-plasma waves is examined through the use of modified reductive perturbation method developed by us [J. Phys. Soc. Jpn. 68 (1999) 1833]. In the analysis it is shown that the lowest order term in the expansion is governed by the nonlinear Schrodinger equation while the second order term is governed by the linear Schrodinger equation. For the small wave number region a set of solution is presented for the evolution equations.Yayın Waves in fluid-filled elastic tubes with a stenosis: Variable coefficients KdV equations(Elsevier B.V., 2007-05-15) Demiray, HilmiIn the present work, by treating the arteries as thin-walled prestressed elastic tubes with a stenosis and the blood as an inviscid fluid we have studied the propagation of weakly nonlinear waves in such a medium, in the longwave approximation, by employing the reductive perturbation method. The variable coefficients KdV and modified KdV equations are obtained depending on the balance between the nonlinearity and the dispersion. By seeking a localized progressive wave type of solution to these evolution equations, we observed that the wave speeds takes their maximum values at the center of stenosis and gets smaller and smaller as one goes away from the stenosis. Such a result seems to reasonable from the physical point of view.
