A perturbative analysis of the time-envelope approximation in strong Langmuir turbulence

We investigate a nonlinear set of coupled-wave equations describing the inertial regime of the strong Langmuir turbulence, namely

1/omega(2) partial derivative(2)E/partial derivative t(2) - 2i partial derivative E/partial derivative t - Delta E = -nE,

1/c(2) partial derivative(2)n/partial derivative 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/w(2) that vanishes under the so-called time-envelope approximation w(2) --> +infinity. 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) --> 0. 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) -->, +infinity 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.