The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence

Abstract

We consider the initial value problem for the Zakharov equations

$$\begin{gathered} \left( Z \right)\frac{1}{{\lambda ^2 }}n_{tt} - \Delta (n + \left| {\rm E} \right|^2 ) = 0n(x,0) = n_0 (x) \hfill \\ n_t (x,0) = n_1 (x) \hfill \\ iE_t + \Delta E - nE = 0E(x,0) = E_0 (x) \hfill \\ \end{gathered} $$

(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|² E=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.