Resonant interaction

In nonlinear systems a resonant interaction is the interaction of three or more waves, usually but not always of small amplitude. 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. Theoretical work on partial differential equations provides insights into chaos theory; there are curious links to number theory. Resonant interactions allow waves to (elastically) scatter, diffuse or to become unstable. 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. The momentum of a wave is understood to be given by its wave vector ${\displaystyle k}$ and its energy ${\displaystyle \omega }$ follows from the dispersion relation for the system.

For example, for three waves in continuous media, the resonant condition is conventionally written as the requirement that ${\displaystyle k_{1}\pm k_{2}\pm k_{3}=0}$ and also ${\displaystyle \omega _{1}\pm \omega _{2}\pm \omega _{3}=0}$, 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).

Although three-wave systems provide the simplest form of resonant interactions in waves, not all systems have three-wave interactions. For example, the deep-water wave equation, a continuous-media system, does not have a three-wave interaction. The Fermi–Pasta–Ulam–Tsingou problem, a discrete-media system, does not have a three-wave interaction. It does have a four-wave interaction, but this is not enough to thermalize the system; that requires a six-wave interaction. As a result, the eventual thermalization time goes as the inverse eighth power of the coupling—clearly, a very long time for weak coupling—thus allowing the famous FPUT recurrences to dominate on "normal" time scales.

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. In such a case, the system can be expressed in terms of a Hamiltonian, formulated in terms of canonical coordinates ${\displaystyle p,q}$. To avoid notational confusion, write ${\displaystyle \psi ,\phi }$ for these two; they are meant to be conjugate variables satisfying Hamilton's equation. These are to be understood as functions of the configuration space coordinates ${\displaystyle {\vec {x}},t}$, i.e. functions of space and time. Taking the Fourier transform, write

and likewise for ${\displaystyle {\hat {\phi }}({\vec {k}})}$. Here, ${\displaystyle {\vec {k}}}$ is the wave vector. When "on shell", it is related to the angular frequency ${\displaystyle \omega }$ by the dispersion relation. The ladder operators follow in the canonical fashion:

with ${\displaystyle 2f(\omega )}$ some function of the angular frequency. The ${\displaystyle a,a^{*}}$ correspond to the normal modes of the linearized system. The Hamiltonian (the energy) can now be written in terms of these raising and lowering operators (sometimes called the "action density variables") as