The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence

We consider the initial value problem for the Zakharov equations {Mathematical expression} (x∈ℝ^k, k=2, 3, t ≧0) which model the propagation of Langmuir waves in plasmas. For suitable initial data solutions are shown to exist for a time interval independent of λ, a parameter proportional to the ion acoustic speed. For such data, solutions of (Z) converge as λ → ∞ to a solution of the cubic nonlinear Schrödinger equation (CSE)iE_t+ΔE+|E|²E=0. We consider both weak and strong solutions. For the case of strong solutions the results are analogous to previous results on the incompressible limit of compressible fluids.