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IMA Journal of Applied Mathematics Advance Access published online on March 4, 2009
IMA Journal of Applied Mathematics, doi:10.1093/imamat/hxp010
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© The Author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Focusing and defocusing cases of the purely elliptic generalized Davey–Stewartson system

A. Eden

Department of Mathematics, Bogaziçi University, Bebek 34342, Istanbul, Turkey and Turkish Scientific and Technological Research Council, Feza Gürsey Institute, Çengelköy 34684, Istanbul, Turkey

T. B. Gürel{dagger} and E. Kuz{ddagger}

Department of Mathematics, Bogaziçi University, Bebek 34342, Istanbul, Turkey

{dagger} Email: bgurel{at}boun.edu.tr, t.burak.gurel{at}gmail.com

Received on September 23, 2008; Accepted on January 18, 2009

We define the focusing and the defocusing cases for the purely elliptic generalized Davey–Stewartson system. These cases are mutually exclusive and exhaustive and therefore close the gap that was left in the previous studies. In the defocusing case, all solutions exist globally. In the focusing case, any initial data can be scaled to one with negative energy. The solution with the scaled initial data then blows up in finite time. We also show the existence of standing waves and the global existence and scattering of solutions with subminimal mass. Our results equally apply to the elliptic almost-cubic non-linear Schrödinger equation as described in Eden & Kuz (2009, Commun. Pure Appl. Anal.).

Keywords: blow-up; standing waves; NLS; GDS system; scattering.

{ddagger} Present address: Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA.


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