Adiabatic limit of the zakharov-rubenchik equation

The Zakharov-Rubenchik system { i ∂ T B + ω ∂ XX B − k ( u − υ 2 ρ + q | B | 2 ) B = 0 , ∈ ∂ T ρ + ∂ X ( u − υ ρ ) = − k ∂ X | B | 2 , ∈ ∂ T u + ∂ X ( β ρ − υ u ) = k 2 υ ∂ X | B | 2 appears in the context of Alfvén waves propagating in a magnetized plasma. This system “contains” the well-known Zakharov equation and the Benney equation for the interaction of high and low frequency waves. We prove the pointwise convergence of the magnetic field B to a solution of the nonlinear Schrödinger equation, in the adiabatic limit ∈ → 0.