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Listeleniyor 1 - 7 / 7
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    Standing waves for a generalized Davey-Stewartson system
    (IOP Publishing, 2006-10-27) Eden, Osman Alp; Erbay, Saadet
    In this paper, we establish the existence of non-trivial solutions for a semi-linear elliptic partial differential equation with a non-local term. This result allows us to prove the existence of standing wave ( ground state) solutions for a generalized Davey-Stewartson system. A sharp upper bound is also obtained on the size of the initial values for which solutions exist globally.
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    Numerical simulation of blow-up solutions for the generalized Davey-Stewartson system
    (Taylor & Francis Ltd, 2011-03) Muslu, Gülçin Mihriye; Erbay, Hüsnü Ata
    Blow-up solutions for the generalized Davey-Stewartson system are studied numerically by using a split-step Fourier method. The numerical method has spectral-order accuracy in space and first-order accuracy in time. To evaluate the ability of the split-step Fourier method to detect blow-up, numerical simulations are conducted for several test problems, and the numerical results are compared with the analytical results available in the literature. Good agreement between the numerical and analytical results is observed.
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    A numerical study of the long wave-short wave interaction equations
    (Elsevier B.V., 2007-03-07) Borluk, Handan; Muslu, Gülçin Mihriye; Erbay, Hüsnü Ata
    Two numerical methods are presented for the periodic initial-value problem of the long wave-short wave interaction equations describing the interaction between one long longitudinal wave and two short transverse waves propagating in a generalized elastic medium. The first one is the relaxation method, which is implicit with second-order accuracy in both space and time. The second one is the split-step Fourier method, which is of spectral-order accuracy in space. We consider the first-, second- and fourth-order versions of the split-step method, which are first-, second- and fourth-order accurate in time, respectively. The present split-step method profits from the existence of a simple analytical solution for the nonlinear subproblem. We numerically test both the relaxation method and the split-step schemes for a problem concerning the motion of a single solitary wave. We compare the accuracies of the split-step schemes with that of the relaxation method. Assessments of the efficiency of the schemes show that the fourth-order split-step Fourier scheme is the most efficient among the numerical schemes considered.
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    Reducing a generalized Davey-Stewartson system to a non-local nonlinear Schrodinger equation
    (Pergamon-Elsevier Science Ltd, 2009-07-30) Eden, Osman Alp; Erbay, Saadet; Hacınlıyan, Irma
    In the present study, we consider a generalized (2 + 1) Davey-Stewartson (GDS) system consisting of a nonlinear Schrodinger (NLS) type equation for the complex amplitude of a short wave and two asymmetrically coupled linear wave equations for long waves propagating in an infinite elastic medium. We obtain integral representation of solutions to the coupled linear wave equations and reduce the GDS system to a NLS equation with non-local terms. Finally, we present localized solutions to the GDS system, decaying in both spatial coordinates, for a special choice of parameters by using the integral representation of solutions to the coupled linear wave equations.
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    Closing the gap in the purely elliptic generalized Davey-Stewartson system
    (Pergamon-Elsevier Science Ltd, 2008-10-15) Eden, Osman Alp; Erbay, Hüsnü Ata; Muslu, Gülçin Mihriye
    In this note we improve the results presented previously on global existence and global nonexistence for the Solutions of the purely elliptic generalized Davey-Stewartson system. These results left a gap in the parameter range where neither a global existence result nor a global nonexistence result could be established. Here we are able to show that when the coupling parameter is negative there is no gap. Moreover, in the case where the coupling parameter is positive we reduce the size of the gap.
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    Two-dimensional wave packets in an elastic solid with couple stresses
    (Pergamon-Elsevier Science Ltd, 2004-08) Babaoğlu, Ceni; Erbay, Saadet
    The problem of (2+1) (two spatial and one temporal) dimensional wave propagation in a bulk medium composed of an elastic material with couple stresses is considered. The aim is to derive (2+1) non-linear model equations for the description of elastic waves in the far field. Using a multi-scale expansion of quasi-monochromatic wave solutions, it is shown that the modulation of waves is governed by a system of three non-linear evolution equations. These equations involve amplitudes of a short transverse wave, a long transverse wave and a long longitudinal wave, and will be called the "generalized Davey Stewartson equations". Under some restrictions on parameter values, the generalized Davey-Stewartson equations reduce to the Davey-Stewartson and to the non-linear Schrodinger equations. Finally, some special solutions involving sech-tanh-tanh and tanh-tanh-tanh type solitary wave solutions are presented.
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    Stability of solitary waves for three-coupled long wave-short wave interaction equations
    (Oxford Univ Press, 2011-08) Borluk, Handan; Erbay, Saadet
    In this paper, we consider a three-component system of 1D long wave-short wave interaction equations. The system has two-parameter family of solitary wave solutions. We prove orbital stability of the solitary wave solutions using variational methods.