ON THE GROUND-STATES OF VECTOR NONLINEAR SCHRODINGER-EQUATIONS

We consider a class of vector nonlinear Schrodinger equations, which a rise in the infinite ion acoustic speed limit of the Zakharov equation s. We define a ground state as a minimizer of an appropriate energy fu nctional. Ground states satisfy a nonlinear elliptic system of partial differential equations. We show, in certain parameter regimes, that a ground state cannot be a radial vector field (the gradient of a funct ion which depends only on the distance to some fixed origin of coordin ates). This was conjectured and supported by numerical observations of Zakharov et al, ([26], [6]). In a special case, corresponding to a Gi nzburg Landau energy functional, we prove that the ground state is a v ector field whose components are constant multiples of the ground stat e of the analogous scalar variational problem. It follows, in this cas e, that the ground state is essentially unique. This gives a character ization of ground states (or minimum action solutions) constructed by Brezis and Lieb [1].