Genelleştirilmiş Davey-Stewartson (GDS) sistemi, şeklinde verilmiş bir nonlineer kısmi türevli diferansiyel denklem sistemidir. Burada kompleks değerli ve reel değerli fonksiyonlar olup, uzaysal koordinatlarının ve zamanını fonksiyonlarıdır. Ve reel parametrelerdir. Ayrıca, katsayılar arasında şeklinde bir bağıntı sağlanmaktadır. GDS sistemi sonsuz bir genelleştirilmiş elastik ortamda yayılan quazi monokromatik dalgaların modulasyonunu karakterize eden sistem olarak önerilmiştir (Babaoğlu ve Erbay, 2004). Bu çalışmada katsayılarının işaretlerine göre Eliptik Eliptik Eliptik (EEE) ve Hiperbolik Eliptik Eliptik (HEE) durumları göz önüne alınacaktır (Eden vd., 2006). GDS sisteminin en önemli özel halleri Nonlineer Schrödinger (NLS) denklemi ve Davey-Stewartson (DS) sistemidir. NLS denkleminde olduğu gibi, DS sisteminin çözümleri de pseudo-konformal dönüşüm altında invaryant kalmaktadır. DS sisteminin Hiperbolik-Eliptik (HE) durumunda pseudo-konformal dönüşüm yardımıyla bir analitik patlama profili elde edilirken, eliptik NLS denklemi için bu invaryantlık çözümlerin patlama profillerinin anlaşılmasında temel rol oynar. Bu çalışmada GDS sisteminin de çözümlerinin pseudo-konformal dönüşüm altında invaryant kaldığı gösterilmiş ve pseudo-konformal invaryantlık kullanılarak bulunan iki yeni sonuç sunulmuştur. HEE durumda, fiziksel parametreler üzerine bazı koşullar koyarak bir patlama profili elde edilmiştir. Ancak bu koşullar bir özel “radyal” çözümün varlığı için gerekli koşullara dönüşür. EEE durumda, düzgün çözümlerin normunun zamanla cebirsel olarak sıfıra gittiği gösterilmiştir. Anahtar Kelimeler: Pseudo-konformal invaryant, Patlama profili, stabilitesi.The Generalized Davey-Stewartson (GDS) system is given by where and, are, respectively, the complex and the real valued functions of spatial coordinates and the time. The parameters and are real constants and is normalized as. The parametric relation follows from the structure of the physical constants and plays a key role in the analysis of these equations. The GDS system was derived to model dimensional wave propagation in a bulk medium composed of elastic material with couple stresses (Babaoğlu and Erbay, 2004). Four conserved quantities, corresponding to mass, momentum in the and directions and energy, were derived in (Babaoğlu et al., 2004). Furthermore, these equations were classified according to the signs of as Elliptic Elliptic Elliptic (EEE), Elliptic Elliptic Hyperbolic (EEH), Elliptic Hyperbolic Hyperbolic (EHH), Hyperbolic Elliptic Elliptic (HEE), Hyperbolic Hyperbolic Hyperbolic (HHH), Hyperbolic Elliptic Hyperbolic (HEH). Here we will only be considering EEE and HEE cases. It is well-known that many equations can be expressed as a NLS equation with additional and possibly non-local terms (Constantin, 1990). For example, in the elliptic elliptic and hyperbolic elliptic cases of the Davey-Stewartson (DS) system can be written as with, where is a linear pseudo-differential operator with a homogeneous symbol of order zero. The non-local term acts in many ways like a cubic nonlinearity: they have similar scaling properties and the interaction between these two nonlinear terms determine the global behaviour of the solutions. In Babaoğlu et al., (2004), for the HEE and EEE cases the GDS system has been expressed as a NLS equation with additional term. This representation leads one to expect some of properties of the NLS equation to remain valid for the GDS system. In two space dimensions, the solutions of the Schrödinger equations with cubic nonlinearity (NLS) are invariant under the pseudo-conformal transformation. In addition to its inherent interests, this invariance has far reaching consequences leading to a better understanding of the blow-up profiles; global existence of the solutions; as well as their -stability. As it is the case for the NLS equation, the solutions of the DS system are invariant under the pseudo-conformal transformation. For the elliptic NLS, this invariance plays a key role in understanding the blow-up profile of solutions, whereas in the hyperbolic-elliptic case of DS system an explicit blow-up profile is obtained via the pseudo-conformal invariance. The main aim of this study, is to highlight the importance of the pseudo-conformal invariance for the GDS system (Eden et al., 2006). We start by recalling some of the work done for the classification of the GDS system as well as its conserved quantities. Then the solutions are also shown to be invariant under a scale transformation. The corresponding conserved quantity, denoted by, is used for a derivation of the virial identity. Next, the pseudo-conformal transformation for the GDS system is stated and its invariant, denoted by, is found. Then, we focus on the HEE case for the GDS system. Starting from a solution ansatz in the spirit of (Ozawa, 1992), a set of conditions are found on the underlying parameters. These conditions also turn out to be necessary conditions for the existence of a "radial" steady state solution. The pseudo-conformal transformation converts this steady-state solution into a time dependent one which blows-up in finite time. Finally, following an idea given in (Weinstein, 1989) we show that for, the norms of smooth solutions of the GDS system in the EEE case converge to as. Keywords: Pseudo-conformal invariance, Blow-up profile, -stability
