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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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    Global existence and nonexistence results for a generalized Davey-Stewartson system
    (IOP Publishing, 2004-12-03) Babaoğlu, Ceni; Eden, Osman Alp; Erbay, Saadet
    We 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.
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    On travelling wave solutions of a generalized Davey-Stewartson system
    (Oxford Univ Press, 2005-02) Eden, Osman Alp; Erbay, Saadet
    The generalized Davey-Stewartson (GDS) equations, as derived by Babaoglu & Erbay (2004, Int. J. Non-Linear Mech., 39, 941-949), is a system of three coupled equations in (2 + 1) dimensions modelling wave propagation in an infinite elastic medium. The physical parameters (gamma, m(1), m(2), lambda and n) of the system allow one to classify the equations as elliptic-elliptic-elliptic (EEE), elliptic-elliptic-hyperbolic (EEH), elliptic-hyperbolic-hyperbolic (EHH), hyperbolic-elliptic-elliptic (HEE), hyperbolic-hyperbolic-hyperbolic (HHH) and hyperbolic-elliptic-hyperbolic (HEH) (Babaoglu et al., 2004, preprint). In this note, we only consider the EEE and HEE cases and seek travelling wave solutions to GDS systems. By deriving Pohozaev-type identities we establish some necessary conditions on the parameters for the existence of travelling waves, when solutions satisfy some integrability conditions. Using the explicit solutions given in Babaoglu & Erbay (2004) we also show that the parameter constraints must be weaker in the absence of such integrability conditions.
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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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    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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    Two remarks on a generalized Davey-Stewartson system
    (Elsevier Ltd, 2006-03-01) Eden, Osman Alp; Erbay, Hüsnü Ata; Muslu, Gülçin Mihriye
    We present two results on a generalized Davey-Stewartson system, both following from the pseudo-conformal invariance of its solutions. In the hyperbolic-elliptic-elliptic case, under some conditions on the physical parameters, we establish a blow-up profile. These conditions turn out to be necessary conditions for the existence of a special '' radial '' solution. In the elliptic-elliptic-elliptic case, under milder conditions, we show the L-P-norms of the solutions decay to zero algebraically in time for 2 < p < infinity.