Tachyonic and parametric resonances for massive particle production in an intense plane wave background

We investigate the stability of an intensive plane wave of a massless or light field $\phi$ in a trilinear scalar model $g\phi\chi^2$ due to the resonant production of massive $\chi$ particles in a perturbatively forbidden regime. We apply two methods: first, we solve the Heisenberg equation for the quantum amplitudes of the field $\chi$ in an external plane wave background, generalizing the solution of A.Arza. Second, for the light but massive $\phi$ we perform the relativistic boost to the rest frame of $\phi$, reducing the problem to the stability of the thoroughly investigated massive condensate. It turns out that the stability properties are significantly different for the cases of massless and light fields. In the first case, one should adopt the Heisenberg equation approach, since the alternative method cannot provide a comprehensive outcome. In the second case, the use of the Mathieu equation provides a more accurate solution, while for the massless case, this approach is not applicable.