The Matrix Nonlinear Schr¨odinger Equation in Dimension 2

In this paper we study the existence of global solutions to the Cauchy problem for the matrix nonlinear Schr¨odinger equationŽMNLS.in 2 space dimensions. A sharp condition for the global existence is obtained for this equation. This condition is in terms of an exact stationary solution of a semilinear elliptic equation. In the scalar case, the MNLS reduces to the well-known cubic nonlinear Schr¨odinger equation for which existence of solutions has been studied by many authors.