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On a nonlocal Zakharov equation. (English) Zbl 0830.35123
This article deals with some equations introduced by Zakharov to describe Langmuir plasma turbulence: \[ i \dot \varphi + \Delta \varphi = - B (n \varphi), \quad \lambda^{- 2} \ddot n - \Delta n = \Delta |\varphi |^2, \] where \(B = \nabla \Delta^{-1} \nabla\). The initial value problem \(\varphi (x,0) = \varphi_0 (x),\) \(n (x,0) = n_0 (x),\) \(n_t (x,0) = n_1(x)\) is investigated. The existence and the uniqueness for the Cauchy problem is proved. Moreover, the case when \(\lambda\) tends to \(\infty\) is studied.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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