Stability of Solitary Waves for a Generalized Nonlinear Coupled Schrodinger Systems

In this paper we show that the standing waves of the form (eu(x), eu(x)), β > 0, u(x) real and positive, are stable for the system i ∂u ∂t + uxx + (|u| + γ|v||u|)u = 0 i ∂v ∂t + vxx + (γ|u||v| + |v|)v = 0 provided 2 ≤ p < 3 and 0 < γ 6= p− 1. The Morse index of such solution is one for γ > p − 1 and two for 0 < γ < p− 1 but it is stable in both cases.