Resonant interaction

Last updated April 17, 2024

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.^[1] 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 $k$ and its energy $\omega$ 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 $k_{1}\pm k_{2}\pm k_{3}=0$ and also $\omega _{1}\pm \omega _{2}\pm \omega _{3}=0$ , 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.^[2] 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.^[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.

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.^[4]^[2] In such a case, the system can be expressed in terms of a Hamiltonian, formulated in terms of canonical coordinates $p,q$ . To avoid notational confusion, write $\psi ,\phi$ 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 ${\vec {x}},t$ , i.e. functions of space and time. Taking the Fourier transform, write

{\hat {\psi }}({\vec {k}})=\int e^{-i{\vec {k}}\cdot {\vec {x}}}\;\psi ({\vec {x}})\;dx

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

{\hat {\phi }}({\vec {k}})={\sqrt {2f(\omega )}}\;\;\left(a_{k}+a_{-k}^{*}\right)

{\hat {\psi }}({\vec {k}})=-i{\sqrt {\frac {2}{f(\omega )}}}\;\;\left(a_{k}-a_{-k}^{*}\right)

with $2f(\omega )$ some function of the angular frequency. The $a,a^{*}$ 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

H=H_{0}(a,a^{*})+\epsilon H_{1}(a,a^{*})

Here, the first term $H_{0}(a,a^{*})$ is quadratic in $a,a^{*}$ and represents the linearized theory, while the non-linearities are captured in $H_{1}(a,a^{*})$ , 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 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 $a\to a^{\prime }=a+{\mathcal {O}}(\epsilon )$ and likewise for $a^{*}$ , and rewrites the system in terms of these new, "free" (or at least, freer) modes. Properly done, this leaves $H_{1}$ expressed only with terms that are resonantly interacting. If $H_{1}$ 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,^[5] 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 $a_{1}$ below, the time evolution has the generic form

{\frac {\partial a_{1}}{\partial t}}+i\omega _{1}=-i\int dk_{2}\cdots dk_{n}\;T_{1\cdots n}\;a_{2}^{\pm }\cdots a_{n}^{\pm }\;\delta _{1\pm 2\pm \cdots \pm n}

with $T_{1\cdots n}$ the transfer coefficients for the n-wave interaction, and the $\delta _{1\pm 2\pm \cdots \pm n}=\delta (k_{1}\pm k_{2}\pm \cdots \pm k_{n})$ capturing the notion of the conservation of energy/momentum implied by the resonant interaction. Here $a_{k}^{\pm }$ is either $a$ or $a^{*}$ 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.

In deep water, there are no three-wave interactions between surface 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] The Hamiltonian formalism for deep water waves was given by Zakharov 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 the jet-stream and oceanic waves that move along the thermocline. 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 to Diophantine equations, normally considered to be a topic in number theory.^[9] Constructive methods for solving Diophantine 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 large attenuation. Resonant interaction between acoustic waves and soliton internal waves have been proposed as the source of these anomalies.^[11]
In astrophysics, non-linear resonant interactions between warping and oscillations in the relativistically spinning accretion disk around a black hole have been proposed as the origin of observed kilohertz quasi-periodic oscillations in low-mass x-ray binaries.^[12] The non-linearity providing the coupling is due to general 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).
During spacecraft atmospheric entry, the high speed of the spacecraft heats air to a red-hot plasma. 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 of electron microscopes to the high temporal resolution of lasers, 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 the Klein–Gordon equation can be accelerated by gravitational waves (e.g. those emitted from black hole mergers.)^[16]
The physical basis for macromolecular bioactivity — molecular recognition — the protein-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 "just hydrogen bonds"). The Informational Spectrum Method (ISM) describes such molecular binding in terms of resonant interactions.^[17]^[18] Given a protein, the valence electrons on various amino acids delocalize, and have some freedom of movement within the protein. Their behavior can be modelled in a relatively straightforward way with an electron-ion pseudopotential (EIIP), one for each distinct amino acid or nucleotide. The result of modelling provides spectra, which can be accessed experimentally, thus confirming numerical results. In addition, the model provides the needed dispersion relation from which the resonant interactions can be deduced. Resonant interactions are obtained by computing cross-spectra. Since resonant interactions mix states (and thus alter entropy), recognition might proceed through entropic forces.
Resonant interaction between high-frequency electromagnetic fields and cancer cells has been proposed as a method for treating cancer.^[19]

Related Research Articles

<span class="mw-page-title-main">Nonlinear optics</span> Branch of physics

Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (when the electric field of the light is >10⁸ V/m and thus comparable to the atomic electric field of ~10¹¹ V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves.

In fluid dynamics, the baroclinity of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid. In meteorology a baroclinic flow is one in which the density depends on both temperature and pressure. A simpler case, barotropic flow, allows for density dependence only on pressure, so that the curl of the pressure-gradient force vanishes.

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic, cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

In condensed matter physics, a spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry. From the equivalent quasiparticle point of view, spin waves are known as magnons, which are bosonic modes of the spin lattice that correspond roughly to the phonon excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a ferromagnet's spontaneous magnetization. The energies of spin waves are typically only $μeV$ in keeping with typical Curie points at room temperature and below.

In physics, the Fermi–Pasta–Ulam–Tsingou (FPUT) problem or formerly the Fermi–Pasta–Ulam problem was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactly periodic behavior – called Fermi–Pasta–Ulam–Tsingou recurrence – instead of the expected ergodic behavior. This came as a surprise, as Enrico Fermi, certainly, expected the system to thermalize in a fairly short time. That is, it was expected for all vibrational modes to eventually appear with equal strength, as per the equipartition theorem, or, more generally, the ergodic hypothesis. Yet here was a system that appeared to evade the ergodic hypothesis. Although the recurrence is easily observed, it eventually became apparent that over much, much longer time periods, the system does eventually thermalize. Multiple competing theories have been proposed to explain the behavior of the system, and it remains a topic of active research.

The Jaynes–Cummings model is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.

In spectroscopy, the Autler–Townes effect, is a dynamical Stark effect corresponding to the case when an oscillating electric field is tuned in resonance to the transition frequency of a given spectral line, and resulting in a change of the shape of the absorption/emission spectra of that spectral line. The AC Stark effect was discovered in 1955 by American physicists Stanley Autler and Charles Townes.

Inertial waves, also known as inertial oscillations, are a type of mechanical wave possible in rotating fluids. Unlike surface gravity waves commonly seen at the beach or in the bathtub, inertial waves flow through the interior of the fluid, not at the surface. Like any other kind of wave, an inertial wave is caused by a restoring force and characterized by its wavelength and frequency. Because the restoring force for inertial waves is the Coriolis force, their wavelengths and frequencies are related in a peculiar way. Inertial waves are transverse. Most commonly they are observed in atmospheres, oceans, lakes, and laboratory experiments. Rossby waves, geostrophic currents, and geostrophic winds are examples of inertial waves. Inertial waves are also likely to exist in the molten core of the rotating Earth.

Resonance fluorescence is the process in which a two-level atom system interacts with the quantum electromagnetic field if the field is driven at a frequency near to the natural frequency of the atom.

In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure and optical properties of crystalline solids. It is pronounced "k dot p", and is also called the "k·p method". This theory has been applied specifically in the framework of the Luttinger–Kohn model, and of the Kane model.

The light-front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where $plays the role of time and the corresponding spatial coordinate is . Here, is the ordinary time, is one Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and, are untouched and often called transverse or perpendicular, denoted by symbols of the type . The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others.$

In physics, nonlinear resonance is the occurrence of resonance in a nonlinear system. In nonlinear resonance the system behaviour – resonance frequencies and modes – depends on the amplitude of the oscillations, while for linear systems this is independent of amplitude. The mixing of modes in non-linear systems is termed resonant interaction.

<span class="mw-page-title-main">Surface plasmon polariton</span> Electromagnetic waves that travel along an interface

Surface plasmon polaritons (SPPs) are electromagnetic waves that travel along a metal–dielectric or metal–air interface, practically in the infrared or visible-frequency. The term "surface plasmon polariton" explains that the wave involves both charge motion in the metal and electromagnetic waves in the air or dielectric ("polariton").

Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is thermal energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is different made (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.

The semiconductor Bloch equations describe the optical response of semiconductors excited by coherent classical light sources, such as lasers. They are based on a full quantum theory, and form a closed set of integro-differential equations for the quantum dynamics of microscopic polarization and charge carrier distribution. The SBEs are named after the structural analogy to the optical Bloch equations that describe the excitation dynamics in a two-level atom interacting with a classical electromagnetic field. As the major complication beyond the atomic approach, the SBEs must address the many-body interactions resulting from Coulomb force among charges and the coupling among lattice vibrations and electrons.

In quantum optics, a superradiant phase transition is a phase transition that occurs in a collection of fluorescent emitters, between a state containing few electromagnetic excitations and a superradiant state with many electromagnetic excitations trapped inside the emitters. The superradiant state is made thermodynamically favorable by having strong, coherent interactions between the emitters.

Hopfield dielectric – in quantum mechanics a model of dielectric consisting of quantum harmonic oscillators interacting with the modes of the quantum electromagnetic field. The collective interaction of the charge polarization modes with the vacuum excitations, photons leads to the perturbation of both the linear dispersion relation of photons and constant dispersion of charge waves by the avoided crossing between the two dispersion lines of polaritons. Similarly to the acoustic and the optical phonons and far from the resonance one branch is photon-like while the other charge wave-like. Mathematically the Hopfield dielectric for the one mode of excitation is equivalent to the Trojan wave packet in the harmonic approximation. The Hopfield model of the dielectric predicts the existence of eternal trapped frozen photons similar to the Hawking radiation inside the matter with the density proportional to the strength of the matter-field coupling.

In the physics of continuous media, spatial dispersion is usually described as a phenomenon where material parameters such as permittivity or conductivity have dependence on wavevector. Normally, such a dependence is assumed to be absent for simplicity, however spatial dispersion exists to varying degrees in all materials.

In nonlinear systems, the three-wave equations, sometimes called the three-wave resonant interaction equations or triad resonances, describe small-amplitude waves in a variety of non-linear media, including electrical circuits and non-linear optics. They are a set of completely integrable nonlinear partial differential equations. Because they provide the simplest, most direct example of a resonant interaction, have broad applicability in the sciences, and are completely integrable, they have been intensively studied since the 1970s.

References

↑ McComas, C. Henry; Bretherton, Francis P. (1977). "Resonant interaction of oceanic internal waves". Journal of Geophysical Research. 82 (9): 1397–1412. Bibcode:1977JGR....82.1397M. doi:10.1029/JC082i009p01397.
1 2 3 Janssen, P. A. E. M. (2009). "On some consequences of the canonical transformation in the hamiltonian theory of water waves". J. Fluid Mech. 637: 1–44. Bibcode:2009JFM...637....1J. doi:10.1017/S0022112009008131.
1 2 Onorato, Miguel; Vozella, Lara; Proment, Davide; Lvov, Yuri V. (2015). "A route to thermalization in the α-Fermi–Pasta–Ulam system". Proc. Natl. Acad. Sci. U.S.A. 112 (14): 4208–4213. arXiv: 1402.1603 . Bibcode:2015PNAS..112.4208O. doi:10.1073/pnas.1404397112. PMC 4394280 . PMID 25805822.
1 2 Zakharov, V. (1968). "Stability of periodic waves of finite amplitude on a surface of a deep fluid". J. Appl. Mech. Tech. Phys. 9 (2): 190–194. Bibcode:1968JAMTP...9..190Z. doi:10.1007/BF00913182. S2CID 55755251.
↑ "Small denominators", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
↑ Bonnefoy, F.; Haudin, F.; Michel, G.; Semin, B.; Humbert, T.; Aumaître, S.; Berhanu, M.; Falcon, E. (2018). "Observation of resonant interactions among surface gravity waves". J. Fluid Mech. 805: R3. arXiv: 1606.09009 . doi:10.1017/jfm.2016.576. S2CID 1694066.
↑ Yang, Bo; Yang, Jianke (2020). "General rogue waves in the three-wave resonant interaction systems". arXiv: 2005.10847 [nlin.SI].
↑ Longuet-Higgins, Michael Selwyn; Gill, Adrian Edmund (1967). "Resonant interactions between planetary waves". Proceedings of the Royal Society A. 299 (1456): 120–144. Bibcode:1967RSPSA.299..120L. doi:10.1098/rspa.1967.0126. S2CID 120179723.
↑ Kishimoto, Nobu; Yoneda, Tsuyoshi (2017). "A number theoretical observation of a resonant interaction of Rossby waves". Kodai Mathematical Journal. 40 (1): 16–20. arXiv: 1409.1031 . doi:10.2996/kmj/1490083220. S2CID 118262278.
↑ Kartashova, Elena (2010), Nonlinear Resonance Analysis. Theory, Computation, Applications, Cambridge University Press. https://www.cambridge.org/core/books/nonlinear-resonance-analysis/76D57DB3D4FDB65B9505E926FE88B99E
↑ Zhou, Ji‐xun; Zhang, Xue‐zhen (1991). "Resonant interaction of sound wave with internal solitons in the coastal zone". The Journal of the Acoustical Society of America. 90 (4): 2042–2054. Bibcode:1991ASAJ...90.2042Z. doi:10.1121/1.401632.
↑ Kato, Shoji (2004). "Wave–Warp Resonant Interactions in Relativistic Disks and kHz QPOs". Publications of the Astronomical Society of Japan. 56 (3): 599–607. Bibcode:2004PASJ...56..559K. doi: 10.1093/pasj/56.3.599 .
↑ Bogatskaya, A. V.; Klenov, N. V.; Tereshonok, M. V.; Adjemov, S. S.; Popov, A. M. (2018). "Resonant interaction of electromagnetic wave with plasma layer and overcoming the radiocommunication blackout problem". Journal of Physics D: Applied Physics. 51 (18): 185602. Bibcode:2018JPhD...51r5602B. doi:10.1088/1361-6463/aab756.
↑ Gover, Avraham; Yariv, Amnon (2020). "Free-Electron–Bound-Electron Resonant Interaction". Physical Review Letters. 124 (6): 064801. Bibcode:2020PhRvL.124f4801G. doi:10.1103/PhysRevLett.124.064801. PMID 32109105.
↑ Vasiliev, A. A.; Artemyev, A. V.; Neishtadt, A. I.; Vainchtein, D. L.; Zelenyi, L. M. (2012). "Resonant Interaction of Charged Particles with Electromagnetic Waves". Chaos, Complexity and Transport. pp. 16–23. doi:10.1142/9789814405645_0002.
↑ Asenjo, Felipe A.; Mahajan, Swadesh M. (2020). "Resonant interaction between dispersive gravitational waves and scalar massive particles". Phys. Rev. D. 101 (6): 063010. Bibcode:2020PhRvD.101f3010A. doi: 10.1103/PhysRevD.101.063010 .
↑ Veljkovic, Veljko (1985). "Is it possible to analyze DNA and protein sequences by the methods of digital signal processing?". IEEE Trans. Biomed. Eng. 32 (5): 337–41. doi:10.1109/TBME.1985.325549. PMID 2581884. S2CID 23892544.
↑ Cosic, Irena (1997). The Resonant Recognition Model of Macromolecular Bioactivity. Berlin: Birkhäuser. ISBN 3-7643-5487-9.
↑ Calabrò, Emanuele; Magazù, Salvatore (2018). "Resonant interaction between electromagnetic fields and proteins: A possible starting point for the treatment of cancer". Electromagnetic Biology and Medicine. 37 (2): 1–14. doi:10.1080/15368378.2018.1499031. PMID 30019948. S2CID 51678917.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] McComas, C. Henry; Bretherton, Francis P. (1977). "Resonant interaction of oceanic internal waves". Journal of Geophysical Research. 82 (9): 1397–1412. Bibcode:1977JGR....82.1397M. doi:10.1029/JC082i009p01397.

[janssen-2] 1 2 3 Janssen, P. A. E. M. (2009). "On some consequences of the canonical transformation in the hamiltonian theory of water waves". J. Fluid Mech. 637: 1–44. Bibcode:2009JFM...637....1J. doi:10.1017/S0022112009008131.

[fput-3] 1 2 Onorato, Miguel; Vozella, Lara; Proment, Davide; Lvov, Yuri V. (2015). "A route to thermalization in the α-Fermi–Pasta–Ulam system". Proc. Natl. Acad. Sci. U.S.A. 112 (14): 4208–4213. arXiv: 1402.1603 . Bibcode:2015PNAS..112.4208O. doi:10.1073/pnas.1404397112. PMC 4394280 . PMID 25805822.

[zakharov-4] 1 2 Zakharov, V. (1968). "Stability of periodic waves of finite amplitude on a surface of a deep fluid". J. Appl. Mech. Tech. Phys. 9 (2): 190–194. Bibcode:1968JAMTP...9..190Z. doi:10.1007/BF00913182. S2CID 55755251.

[5] "Small denominators", Encyclopedia of Mathematics , EMS Press, 2001 [1994]

[6] Bonnefoy, F.; Haudin, F.; Michel, G.; Semin, B.; Humbert, T.; Aumaître, S.; Berhanu, M.; Falcon, E. (2018). "Observation of resonant interactions among surface gravity waves". J. Fluid Mech. 805: R3. arXiv: 1606.09009 . doi:10.1017/jfm.2016.576. S2CID 1694066.

[7] Yang, Bo; Yang, Jianke (2020). "General rogue waves in the three-wave resonant interaction systems". arXiv: 2005.10847 [nlin.SI].

[8] Longuet-Higgins, Michael Selwyn; Gill, Adrian Edmund (1967). "Resonant interactions between planetary waves". Proceedings of the Royal Society A. 299 (1456): 120–144. Bibcode:1967RSPSA.299..120L. doi:10.1098/rspa.1967.0126. S2CID 120179723.

[9] Kishimoto, Nobu; Yoneda, Tsuyoshi (2017). "A number theoretical observation of a resonant interaction of Rossby waves". Kodai Mathematical Journal. 40 (1): 16–20. arXiv: 1409.1031 . doi:10.2996/kmj/1490083220. S2CID 118262278.

[Kartashova2010-10] Kartashova, Elena (2010), Nonlinear Resonance Analysis. Theory, Computation, Applications, Cambridge University Press. https://www.cambridge.org/core/books/nonlinear-resonance-analysis/76D57DB3D4FDB65B9505E926FE88B99E

[11] Zhou, Ji‐xun; Zhang, Xue‐zhen (1991). "Resonant interaction of sound wave with internal solitons in the coastal zone". The Journal of the Acoustical Society of America. 90 (4): 2042–2054. Bibcode:1991ASAJ...90.2042Z. doi:10.1121/1.401632.

[12] Kato, Shoji (2004). "Wave–Warp Resonant Interactions in Relativistic Disks and kHz QPOs". Publications of the Astronomical Society of Japan. 56 (3): 599–607. Bibcode:2004PASJ...56..559K. doi: 10.1093/pasj/56.3.599 .

[13] Bogatskaya, A. V.; Klenov, N. V.; Tereshonok, M. V.; Adjemov, S. S.; Popov, A. M. (2018). "Resonant interaction of electromagnetic wave with plasma layer and overcoming the radiocommunication blackout problem". Journal of Physics D: Applied Physics. 51 (18): 185602. Bibcode:2018JPhD...51r5602B. doi:10.1088/1361-6463/aab756.

[14] Gover, Avraham; Yariv, Amnon (2020). "Free-Electron–Bound-Electron Resonant Interaction". Physical Review Letters. 124 (6): 064801. Bibcode:2020PhRvL.124f4801G. doi:10.1103/PhysRevLett.124.064801. PMID 32109105.

[15] Vasiliev, A. A.; Artemyev, A. V.; Neishtadt, A. I.; Vainchtein, D. L.; Zelenyi, L. M. (2012). "Resonant Interaction of Charged Particles with Electromagnetic Waves". Chaos, Complexity and Transport. pp. 16–23. doi:10.1142/9789814405645_0002.

[16] Asenjo, Felipe A.; Mahajan, Swadesh M. (2020). "Resonant interaction between dispersive gravitational waves and scalar massive particles". Phys. Rev. D. 101 (6): 063010. Bibcode:2020PhRvD.101f3010A. doi: 10.1103/PhysRevD.101.063010 .

[17] Veljkovic, Veljko (1985). "Is it possible to analyze DNA and protein sequences by the methods of digital signal processing?". IEEE Trans. Biomed. Eng. 32 (5): 337–41. doi:10.1109/TBME.1985.325549. PMID 2581884. S2CID 23892544.

[18] Cosic, Irena (1997). The Resonant Recognition Model of Macromolecular Bioactivity. Berlin: Birkhäuser. ISBN 3-7643-5487-9.

[19] Calabrò, Emanuele; Magazù, Salvatore (2018). "Resonant interaction between electromagnetic fields and proteins: A possible starting point for the treatment of cancer". Electromagnetic Biology and Medicine. 37 (2): 1–14. doi:10.1080/15368378.2018.1499031. PMID 30019948. S2CID 51678917.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]