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Yayın Standing waves for a generalized Davey-Stewartson system(IOP Publishing, 2006-10-27) Eden, Osman Alp; Erbay, SaadetIn 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.Yayın Coupled quintic nonlinear Schrodinger equations in a generalized elastic solid(IOP Publishing Ltd, 2004-10-08) Hacınlıyan, Irma; Erbay, SaadetIn the present study, the nonlinear modulation of transverse waves propagating in a cubically nonlinear dispersive elastic medium is studied using a multiscale expansion of wave solutions. It is found that the propagation of quasimonochromatic transverse waves is described by a pair of coupled nonlinear Schrodinger (CNLS) equations. In the process of deriving the amplitude equations, it is observed that for a specific choice of material constants and wavenumber, the coefficient of nonlinear terms becomes zero, and the CNLS equations are no longer valid for describing the behaviour of transverse waves. In order to balance the nonlinear effects with the dispersive effects, by intensifying the nonlinearity, a new perturbation expansion is used near the critical wavenumber. It is found that the long time behaviour of the transverse waves about the critical wavenumber is given by a pair of coupled quintic nonlinear Schrodinger (CQNLS) equations. In the absence of one of the transverse waves, the CQNLS equations reduce to the single quintic nonlinear Schrodinger (QNLS) equation which has already been obtained in the context of water waves. By using a modified form of the so-called tanh method, some travelling wave solutions of the CQNLS equations are presented.Yayın Global existence and nonexistence results for a generalized Davey-Stewartson system(IOP Publishing, 2004-12-03) Babaoğlu, Ceni; Eden, Osman Alp; Erbay, SaadetWe consider a system of three equations, which will be called generalized Davey-Stewartson equations, involving three coupled equations, two for the long waves and one for the short wave propagating in an infinite elastic medium. We classify the system according to the signs of the parameters. Conserved quantities related to mass, momentum and energy are derived as well as a specific instance of the so-called virial theorem. Using these conservation laws and the virial theorem both global existence and nonexistence results are established under different constraints on the parameters in the elliptic-elliptic-elliptic case.
