Innonlinear systems aresonant interaction is the interaction of three or morewaves, usually but not always of small amplitude. Resonant interactions occur when a simple set of criteria couplingwave vectors and thedispersion 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 ofgravity waves, but also finds numerous applications from astrophysics and biology to engineering and medicine. Theoretical work onpartial differential equations provides insights intochaos theory; there are curious links tonumber theory. Resonant interactions allow waves to (elastically)scatter,diffuse or to becomeunstable.^[1] Diffusion processes are responsible for the eventualthermalization of most nonlinear systems; instabilities offer insight intohigh-dimensional chaos andturbulence.

The underlying concept is that when the sum total of the energy and momentum of severalvibrational modes sum to zero, they are free tomix 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 itswave vector${\displaystyle k}$ and its energy${\displaystyle \omega }$ follows from thedispersion relation for the system.

For example, for three waves incontinuous 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 alattice, or in (nonlinear)solid-state systems, the wave vectors are quantized, and thenormal modes can be calledphonons. TheBrillouin 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).

Althoughthree-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.^[2] TheFermi–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.^[3] 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 aHamiltonian formalism. When this is possible, a set of manipulations can be applied, having the form of a generalized, non-linearFourier transform. These manipulations are closely related to theinverse scattering method.

A particularly simple example can be found in the treatment of deep water waves.^[4][2] In such a case, the system can be expressed in terms of a Hamiltonian, formulated in terms ofcanonical coordinates${\displaystyle p,q}$. To avoid notational confusion, write${\displaystyle \psi ,\varphi }$ 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${\displaystyle \overrightarrow{x},t}$,i.e. functions of space and time. Taking theFourier transform, write

${\displaystyle \hat{\psi }(\overrightarrow{k})=\int e^{-i\overrightarrow{k}\cdot \overrightarrow{x}}\psi (\overrightarrow{x})dx}$

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

${\displaystyle \hat{\varphi }(\overrightarrow{k})=\sqrt{2f(\omega )}\left(a_k+a_{-k}^{\ast }\right)}$

${\displaystyle \hat{\psi }(\overrightarrow{k})=-i\sqrt{\frac{2}{f(\omega )}}\left(a_k-a_{-k}^{\ast }\right)}$

with${\displaystyle 2f(\omega )}$ some function of the angular frequency. The${\displaystyle a,a^{\ast }}$ correspond to thenormal 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

${\displaystyle H=H_0(a,a^{\ast })+ϵH_1(a,a^{\ast })}$

Here, the first term${\displaystyle H_0(a,a^{\ast })}$ is quadratic in${\displaystyle a,a^{\ast }}$ and represents the linearized theory, while the non-linearities are captured in${\displaystyle H_1(a,a^{\ast })}$, which is cubic or higher-order.

Given the above as the starting point, the system is then decomposed into "free" and "bound" modes.^[3][2] The bound modes have no independent dynamics of their own; for example, the higher harmonics of asoliton 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${\displaystyle a\to a^{\mathrm{\prime }}=a+\mathcal{O}(ϵ)}$ and likewise for${\displaystyle a^{\ast }}$, and rewrites the system in terms of these new, "free" (or at least, freer) modes. Properly done, this leaves${\displaystyle H_1}$ expressed only with terms that are resonantly interacting. If${\displaystyle H_1}$ is cubic, these are then thethree-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 thesmall divisor problem,^[5] which occurs when there are near-resonances. The terms themselves give the rate or speed of the mixing, and are sometimes calledtransfer coefficients or thetransfer matrix. At the conclusion, one obtains an equation for thetime evolution of the normal modes, corrected by scattering terms. Picking out one of the modes out of the bunch, call it${\displaystyle a_1}$ below, the time evolution has the generic form

${\displaystyle \frac{\mathrm{\partial }{a}_1}{\mathrm{\partial }t}+i{\omega }_1=-i\int d{k}_2\cdots d{k}_n{T}_{1\cdots n}{a}_2^{\pm }\cdots a_n^{\pm }{\delta }_{1\pm 2\pm \cdots \pm n}}$

with${\displaystyle T_{1\cdots n}}$ the transfer coefficients for then-wave interaction, and the${\displaystyle {\delta }_{1\pm 2\pm \cdots \pm n}=\delta (k_1\pm k_2\pm \cdots \pm k_n)}$ capturing the notion of the conservation of energy/momentum implied by the resonant interaction. Here${\displaystyle a_k^{\pm }}$ is either${\displaystyle a}$ or${\displaystyle a^{\ast }}$ as appropriate. For deep-water waves, the above is called theZakharov equation, named afterVladimir E. Zakharov.

Resonant interactions were first considered and described byHenri Poincaré in the 19th century, in the analysis ofperturbation series describing3-bodyplanetary motion. The first-order terms in the perturbative series can be understood for form amatrix; theeigenvalues 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 originalresonant interaction. When in resonance, energy transfer between modes can keep the system in a stablephase-locked state. However, going to second order is challenging in several ways. One is thatdegenerate 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 famoussmall 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.

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.

• In deep water, there are no three-wave interactions betweensurface gravity waves; the shape of the dispersion relation prohibits this. There is, however, a four-wave interaction; it describes the experimentally-observed interaction of obliquely moving waves very well (i.e. with no free parameters or adjustments).^[6] TheHamiltonian formalism for deep water waves was given byZakharov in 1968^[4]
• Rogue waves are unusually large and unexpected oceanic surface waves;solitons are implicated, and specifically, the resonant interactions between three of them.^[7]
• Rossby waves, also known as planetary waves, describe both thejet-stream and oceanic waves that move along thethermocline. There are three-wave resonant interactions of Rossby waves, and so they are commonly studied as such.^[8]
• The resonant interactions of Rossby waves have been observed to have a connection toDiophantine equations, normally considered to be a topic in number theory.^[9] Constructive methods for solvingDiophantine equations appearing in the context of the resonant wave interactions of various types (including Rossby waves) have been first presented by Kartashova in 1990 and can be found in^[10]

• During summertime in shallow coastal waters, low-frequency sound-waves have been observed to propagate in an anomalous fashion. The anomalies are time-dependent,anisotropic, and can exhibit abnormally largeattenuation. Resonant interaction between acoustic waves andsolitoninternal waves have been proposed as the source of these anomalies.^[11]
• Inastrophysics, non-linear resonant interactions between warping and oscillations in the relativistically spinningaccretion disk around ablack hole have been proposed as the origin of observed kilohertzquasi-periodic oscillations in low-massx-ray binaries.^[12] The non-linearity providing the coupling is due togeneral relativity; accretion disks in Newtonian gravity, e.g.Saturn's rings do not have this particular kind of resonant interaction (they do demonstrate many other kinds of resonances, however).
• Duringspacecraftatmospheric entry, the high speed of the spacecraft heats air to a red-hotplasma. This plasma is impenetrable to radio waves, leading to a radio communications blackout. Resonant interactions that mechanically (acoustically) couple the spacecraft to the plasma have been investigated as a means of punching a hole or tunneling out the radiowave, thus re-establishing radio communications during a critical flight phase.^[13]
• Resonant interactions have been proposed as a way of coupling the high spatial resolution ofelectron microscopes to the hightemporal resolution oflasers, allowing precision microscopy in both space and time.^[14] The resonant interaction is between free electrons and bound electrons at the surface of a material.
• Charged particles can be accelerated by resonant interaction with electromagnetic waves.^[15] Scalar particles (neutral atoms) described by theKlein–Gordon equation can be accelerated bygravitational waves (e.g. those emitted from black hole mergers.)^[16]
• The physical basis for macromolecular bioactivity —molecular recognition — theprotein-protein and protein-DNA interaction, is poorly understood. Such interactions are known to be electromagnetic (obviously, its "chemistry"), but are otherwise poorly understood (its not "justhydrogen bonds"). TheInformational Spectrum Method (ISM) describes such molecular binding in terms of resonant interactions.^[17][18] Given a protein, thevalence electrons on variousamino acidsdelocalize, and have some freedom of movement within the protein. Their behavior can be modelled in a relatively straightforward way with an electron-ionpseudopotential (EIIP), one for each distinct amino acid ornucleotide. The result of modelling providesspectra, which can be accessed experimentally, thus confirming numerical results. In addition, the model provides the neededdispersion relation from which the resonant interactions can be deduced. Resonant interactions are obtained by computingcross-spectra. Since resonant interactions mix states (and thus alterentropy), recognition might proceed throughentropic forces.
• Resonant interaction between high-frequency electromagnetic fields andcancer cells has been proposed as a method for treating cancer.^[19]

• Three-wave equation
• Inverse scattering method
• S-matrix
• Orbital resonance
• Nonlinear resonance
• Tidal resonance
• Arnold tongue

• Nonlinear systems

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