The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence

Steven H. Schochet, Michael I. Weinstein

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the initial value problem for the Zakharov equations {Mathematical expression} (x∈ℝk, 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)iEt+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.

Original languageEnglish
Pages (from-to)569-580
Number of pages12
JournalCommunications in Mathematical Physics
Volume106
Issue number4
DOIs
StatePublished - Dec 1986
Externally publishedYes

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