Communications on Pure and Applied Analysis (CPAA)

Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering

Pages: 1803 - 1823, Volume 8, Issue 6, November 2009      doi:10.3934/cpaa.2009.8.1803

       Abstract        Full Text (333.0K)       Related Articles

Alp Eden - Department of Mathematics, Bogaziçi University, Bebek 34342, Istanbul, Turkey (email)
Elİf Kuz - 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 non-local nonlinearities in two space dimensions. This class includes the Davey-Stewartson (DS) equations when the second equation is elliptic and the Generalized Davey-Stewartson (GDS) system when the second and the third equations form an elliptic system. We establish local well-posedness 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 pseudo-conformal 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 blows-up in finite time. We distinguish the focusing and the defocusing cases sharply in terms of a condition on the nonlinearity.

Keywords:  Nonlinear Schrödinger equation, Davey-Stewartson equations, scattering, global existence, focusing solutions.
Mathematics Subject Classification:  Primary: 35G25, 35B40, 35Q55.

Received: October 2008;      Revised: March 2009;      Available Online: August 2009.