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A singular perturbation problem for an envelope equation in plasma physics. (English) Zbl 0884.35097
Summary: We investigate the so-called Langmuir wave envelope approximation, which consists in taking the limit \(\omega\rightarrow\infty\) in the nonlinear plasma wave equation \[ 1/\omega^2\partial^2_tE_\omega-2i\partial_tE_\omega-\Delta E_\omega=f(|E_\omega|^2)E_\omega, \] stated for nonlinearities satisfying \(|f(\rho)|\leq K \rho^\sigma\). For any finite value of \(\omega>1\), the solution \(E_\omega\) with the initial data \(E_\omega(x,t=0) \in H^2(\mathbb{R}^n),\partial_tE_\omega(x,t=0)\in H^1(\mathbb{R}^n)\) is shown to exist locally in time and to be unique. Under some specific conditions including \(\omega\) below a threshold value, we construct solutions \(E_\omega\) that blow up in a finite time with a divergent \(L^2\) norm; nevertheless, in the so-called subcritical case (\(\sigma n <2\)), the solution defined for fixed initial data is global provided that \(\omega\) should be large enough. We demonstrate the strong convergence of \(E_\omega\) towards the nonlinear Schrödinger solution \(E\) reached as \(\omega\rightarrow\infty\), as long as \(E\) exists. In this same limit, we finally discuss the behavior of the time derivative of \(E_\omega\) and compare the blow-up times associated with \(E_\omega\) and with its time-enveloped counterpart \(E\).

35L70 Second-order nonlinear hyperbolic equations
35Q55 NLS equations (nonlinear Schrödinger equations)
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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