Interaction of multiple waves in a nonlinear system
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.
Discussion.
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 formula_1 and its energy formula_2 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 formula_3 and also formula_4, 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.
Hamiltonian formulation.
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 formula_5. To avoid notational confusion, write formula_6 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 formula_7, "i.e." functions of space and time. Taking the Fourier transform, write
formula_8
and likewise for formula_9. Here, formula_10 is the wave vector. When "on shell", it is related to the angular frequency formula_2 by the dispersion relation. The ladder operators follow in the canonical fashion:
formula_12
formula_13
with formula_14 some function of the angular frequency. The formula_15 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
formula_16
Here, the first term formula_17 is quadratic in formula_15 and represents the linearized theory, while the non-linearities are captured in formula_19, which is cubic or higher-order.
Given the above as the starting point, the system is then decomposed into "free" and "bound" modes. The bound modes have no independent dynamics of their own; for example, the higher harmonics of a soliton solution are bound to the fundamental mode, and cannot interact. 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. That is, one re-writes formula_20 and likewise for formula_21, and rewrites the system in terms of these new, "free" (or at least, freer) modes. Properly done, this leaves formula_22 expressed only with terms that are resonantly interacting. If formula_22 is cubic, these are then the three-wave terms; if quartic, these are the four-wave terms, and so on. 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", which occurs when there are near-resonances. The terms themselves give the rate or speed of the mixing, and are sometimes called transfer coefficients or the transfer matrix. At the conclusion, one obtains an equation for the time evolution of the normal modes, corrected by scattering terms. Picking out one of the modes out of the bunch, call it formula_24 below, the time evolution has the generic form
formula_25
with formula_26 the transfer coefficients for the "n"-wave interaction, and the formula_27 capturing the notion of the conservation of energy/momentum implied by the resonant interaction. Here formula_28 is either formula_29 or formula_21 as appropriate. For deep-water waves, the above is called the Zakharov equation, named after Vladimir E. Zakharov.
History.
Resonant interactions were first considered and described by Henri Poincaré in the 19th century, in the analysis of perturbation series describing 3-body planetary motion. 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. However, going to second order is challenging in several ways. One is that degenerate solutions are difficult to diagonalize (there is no unique vector basis for the degenerate space). 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". These can be interpreted as corresponding to chaotic behavior. To roughly summarize, precise resonances lead to scattering and mixing; approximate resonances lead to chaotic behavior.
Applications.
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.