The electrostatic limit for the 3D Zakharov system

We consider the vectorial Zakharov system describing Langmuir waves in a weakly magnetized plasma. In its original derivation (Zakharov, 1972) the evolution for the electric field envelope is governed by a Schrödinger type equation with a singular parameter which is usually large in physical applications. Motivated by this, we study the rigorous limit as this parameter goes to infinity. By using some Strichartz type estimates to control separately the fast and slow dynamics in the problem, we show that the evolution of the electric field envelope is asymptotically constrained onto the space of irrotational vector fields.