A perturbative analysis of the time-envelope approximation in strong Langmuir turbulence
Résumé
We investigate a nonlinear set of coupled-wave equations describing the inertial regime of the strong Langmuir turbulence, namely
$$\frac1{\omega^2}\frac{\partial^2E}{\partial t^2}-2i\frac{\partial E}{\partial t} - \Delta E = -nE,$$
$$\frac1{c^2}\frac{\partial^2n}{\partial t^2}- \Delta n = \Delta|E|^2,$$
which differs from the usual Zakharov equations by the inclusion in the first equation for $E$ of a second time-derivative, multiplied by the parameter $1/\omega^2$ that vanishes under the so-called time-envelope approximation $\omega^2\to+\infty$. From these perturbed Zakharov equations, it is shown that the latter limit is not compatible with a strongly dominant ion inertia corresponding to the formal case $c^2\to0$. In the opposite case, i.e. as $c^2$ remains of order unity, the local-in-time Cauchy problem attached to the above equations is solved and the limit $\omega^2\to+\infty$ is detailed for a fixed value of $c^2$. Under some specific initial data, the solution $E$ is proved to blow up at least in an infinite time provided that $\omega$ lies below a threshold value. When this condition is not fulfilled, the global existence of the solution set $(E, n)$ is finally restored in a one-dimensional space.
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