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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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