Three wave nonlinear interaction: a multimode extension

The slow modulational approximation to the dynamics of high-frequency carrier waves has been proved time and time again as a powerful
technique to deal with system with virtually infinite degrees-of-freedom. Instead of describing the oscillatory modes at their short space and time
scales, the modulational approach allows to obtain approximate governing equations for a smaller and much smoother varying set of dynamical variables:
the amplitudes and phases of the involved waves.

Modulational techniques have been applied to a variety of physical settings, ranging from mechanical waves in solids, to electromagnetic
waves in plasma accelerators. In all cases, the needed condition for accuracy is that the wave interaction is weak enough that amplitudes and
phases indeed change in a much larger space-time scale than the high-frequency times scale and wavelength spatial scale of the carriers.

The modulational theory has been successfully applied to the study of three waves systems, where energy exchange involving three
wave modes is possible if parametric instabilities is present. The wave triplet is a cornerstone in the study of nonlinear wave
interaction and more complex interactive system can be frequently understood with basis on three wave partitions.

Considering the importance of
the three wave interaction, a recent work investigated the behaviour of the triplet dynamics as the coupling grows beyond the proper validity range for
modulational approximations.
It has been found that there exists indeed a critical coupling strength separating modulational and chaotic
regimes, where in the latter amplitudes execute much larger and much faster oscillations than in the former.

In the present work we shall focus on a multimode extension of the triplet interaction to address the behaviour of this multimode system as the nonlinear
coupling between the various modes increase. The question to be examined here is basically whether or not a critical coupling factor is present
defining a transition from a smoother to a less regular type of dynamics, similarly to what happens with an isolated triplet.

As we shall see, a transition will be indeed identified and argued to be of relevance to nonlinear wave fields with cubic nonlinearities in the corresponding
Lagrangian or Hamiltonian functions.