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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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##### References:
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