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A singular perturbation problem in a system of nonlinear Schrödinger equation occurring in Langmuir turbulence. (English) Zbl 0961.76096
Summary: We prove the existence of limit for $$\alpha\to +\infty$$ in the system $$i\partial_t E+\nabla (\nabla\cdot E) -\alpha^2 \nabla\times \nabla \times E=-|E|^{2\sigma}E$$, where $$E:\mathbb{R}^3 \to\mathbb{C}^3$$. This corresponds to an approximation which is made in the context of Langmuir turbulence in plasma physics. The $$L^2$$-subcritical $$\sigma$$ (that is $$\sigma \leq 2/3)$$ and the $$H^1$$-subcritical $$\sigma$$ (that is $$\sigma\leq 2)$$ are studied. In the physical case $$\sigma=1$$, the limit is studied in the $$H^1 (\mathbb{R}^3)$$ norm.

##### MSC:
 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 35Q55 NLS equations (nonlinear Schrödinger equations) 76F99 Turbulence
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##### References:
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