Resonant interactions occur when a simple set of criteria coupling wave vectors and the dispersion equation are met.
The simplicity of the criteria make technique popular in multiple fields.
Its most prominent and well-developed forms appear in the study of gravity waves, but also finds numerous applications from astrophysics and biology to engineering and medicine.
Resonant interactions allow waves to (elastically) scatter, diffuse or to become unstable.
[1] Diffusion processes are responsible for the eventual thermalization of most nonlinear systems; instabilities offer insight into high-dimensional chaos and turbulence.
The underlying concept is that when the sum total of the energy and momentum of several vibrational modes sum to zero, they are free to mix together via nonlinearities in the system under study.
Modes for which the energy and momentum do not sum to zero cannot interact, as this would imply a violation of energy/momentum conservation.
For example, for three waves in continuous media, the resonant condition is conventionally written as the requirement that
, the minus sign being taken depending on how energy is redistributed among the waves.
For waves in discrete media, such as in computer simulations on a lattice, or in (nonlinear) solid-state systems, the wave vectors are quantized, and the normal modes can be called phonons.
The Brillouin zone defines an upper bound on the wave vector, and waves can interact when they sum to integer multiples of the Brillouin vectors (Umklapp scattering).
For example, the deep-water wave equation, a continuous-media system, does not have a three-wave interaction.
In many cases, the system under study can be readily expressed in a Hamiltonian formalism.
When this is possible, a set of manipulations can be applied, having the form of a generalized, non-linear Fourier transform.
These manipulations are closely related to the inverse scattering method.
A particularly simple example can be found in the treatment of deep water waves.
for these two; they are meant to be conjugate variables satisfying Hamilton's equation.
Given the above as the starting point, the system is then decomposed into "free" and "bound" modes.
This can be recognized by the fact that they do not follow the dispersion relation, and have no resonant interactions.
In this case, canonical transformations are applied, with the goal of eliminating terms that are non-interacting, leaving free modes.
, and rewrites the system in terms of these new, "free" (or at least, freer) modes.
Canonical transformations can be repeated to obtain higher-order terms, as long as the lower-order resonant interactions are not damaged, and one skillfully avoids the small divisor problem,[5] which occurs when there are near-resonances.
At the conclusion, one obtains an equation for the time evolution of the normal modes, corrected by scattering terms.
capturing the notion of the conservation of energy/momentum implied by the resonant interaction.
The first-order terms in the perturbative series can be understood for form a matrix; the eigenvalues of the matrix correspond to the fundamental modes in the perturbated solution.
Poincare observed that in many cases, there are integer linear combinations of the eigenvalues that sum to zero; this is the original resonant interaction.
When in resonance, energy transfer between modes can keep the system in a stable phase-locked state.
A second issue is that differences appear in the denominator of the second and higher order terms in the perturbation series; small differences lead to the famous small divisor problem.
Resonant interactions have found broad utility in many areas.
Below is a selected list of some of these, indicating the broad variety of domains to which the ideas have been applied.