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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.

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
Full Text: DOI
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