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The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence. (English) Zbl 0639.76054
Summary: We consider the initial value problem for the Zahkarov equations \[ (1/\lambda^ 2)n_{tt}-\Delta (n+| E|^ 2)=0\quad n(x,0)=n_ 0(x);\quad n_ t(x,0)=n_ 1(x) \] \[ iE_ t+\Delta E-nE=0\quad E(x,0)=E_ 0(x) \] (x\(\in {\mathbb{R}}^ k,\) \(k=2,3,t\geq 0)\) which model the propagation of Langmuir waves in plasmas. For suitable initial data solutions are shown to exist for a time interval dependent of \(\lambda\), a parameter proportional to the ion acoustic speed. For such data, solutions of (Z) converge as \(\lambda\to \infty\) to a solution of the cubic nonlinear Schrödinger equation \(iE_ t+\Delta E+| E|\) \(2E=0\). 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.

MSC:
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35Q99 Partial differential equations of mathematical physics and other areas of application
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