Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering
Alp Eden  Department of Mathematics, Bogaziçi University, Bebek 34342, Istanbul, Turkey (email) Abstract: We give a unified treatment for a class of nonlinear Schrödinger (NLS) equations with nonlocal nonlinearities in two space dimensions. This class includes the DaveyStewartson (DS) equations when the second equation is elliptic and the Generalized DaveyStewartson (GDS) system when the second and the third equations form an elliptic system. We establish local wellposedness of the Cauchy problem in $L^2(\mathbb{R}^2)$, $H^1(\mathbb{R}^2)$, $H^2(\mathbb{R}^2)$ and in $\Sigma=H^1(\mathbb{R}^2)\cap L^2(x^2 dx)$. We show that the maximal interval of existence of solutions in all of these spaces coincides. Then we show that the mass is conserved for $L^2(\mathbb{R}^2)$solutions. Similarly, the energy and the momenta are conserved for the solutions in $H^1(\mathbb{R}^2)$. For the solutions in $\Sigma$, we show that the virial identity and the pseudoconformal conservation hold. We then discuss the global existence and the scattering of solutions when t he underlying Schrödinger equation is of elliptic type. We achieve these results in either of the following three cases: when the initial data is with small enough mass, when an initial data is with subminimal mass and for any initial data in $\Sigma$ in the defocusing case. In the focusing case, we show that when the initial energy of the solution in $\Sigma$ is negative then this solution blowsup in finite time. We distinguish the focusing and the defocusing cases sharply in terms of a condition on the nonlinearity.
Keywords: Nonlinear Schrödinger equation, DaveyStewartson equations, scattering,
global existence, focusing solutions.
Received: October 2008; Revised: March 2009; Available Online: August 2009. 
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