Existence and stability of ground-state solutions of a Schr¨odinger-KdV system John Albert Department of Mathematics University of Oklahoma Norman, OK 73019 Jaime Angulo Pava Department of Mathematics IMECC-UNICAMP C.P. 6065. CEP 13083-970 Campinas S˜ao Paulo, Brazil We consider the coupled Schr¨odinger-KdV system  i(u t + c 1 u x )+ δ 1 u xx = αuv v t + c 2 v x + δ 2 v xxx + γ(v 2 ) x = β(|u| 2 ) x , which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Ground states of the system are, by definition, minimizers of the energy functional subject to constraints on conserved functionals associated with symmetries of the system. In particular, ground states have a simple time dependence because they propagate via those symmetries. For a range of values of the parameters α,β,γ,δ i ,c i , we prove the existence and stability of a two-parameter family of ground states associated with a two-parameter family of symmetries. 1. Introduction In this paper we prove existence and stability results for ground-state solutions to the system of equations  i(u t + c 1 u x )+ δ 1 u xx = αuv v t + c 2 v x + δ 2 v xxx + γ (v 2 ) x = β(|u| 2 ) x , (1.1) where u is a complex-valued function of the real variables x and t, v is a real-valued function of x and t, and the constants c i ,δ i ,α,β,γ are real. We consider here only the pure initial-value problem for (1.1), in which initial data (u(x, 0),v(x, 0)) = (u 0 (x),v 0 (x)) is posed for −∞ <x< ∞, and a solution (u(x, t),v(x, t) is sought for −∞ <x< ∞ and t ≥ 0. Well-posedness results for the pure initial-value problem for (1.1) and certain of its variants have appeared in [7,21,34]; we cite below in Section 5 the specific results we will need here. Systems of the form (1.1) appear as models for interactions between long and short waves in a variety of physical settings. For example, Kawahara et al. [23] de- rived (1.1) as a model for the interaction between long gravity waves and capillary 1