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Three-wave equation

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Nonlinear_systems" class="mw-redirect" title="Nonlinear systems">nonlinear systems</a>, the three-wave equations, sometimes called the three-wave resonant interaction equations or triad resonances, describe small-amplitude <a href="Wave" title="Wave">waves</a> in a variety of <a href="Nonlinear_system" title="Nonlinear system">nonlinear media</a>, including <a href="Water_wave" class="mw-redirect" title="Water wave">water waves</a> in shallow water, <a href="Capillary_wave" title="Capillary wave">capillary waves</a>, the coupling of <a href="Acoustic_wave" title="Acoustic wave">acoustic waves</a> in the <a href="Littoral_zone" title="Littoral zone">littoral zone</a>, acoustic waves in <a href="Plasma_(physics)" title="Plasma (physics)">plasma</a>, oscillations in <a href="Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuits</a> and in <a href="Non-linear_optics" class="mw-redirect" title="Non-linear optics">non-linear optics</a>. They are a set of three <a href="Completely_integrable" class="mw-redirect" title="Completely integrable">completely integrable</a> nonlinear <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a>.

The three-wave equations represent a fundamental deterministic model underlying <a href="Wave_turbulence" title="Wave turbulence">wave turbulence</a> theory and serve as a paradigmatic example of <a href="Resonant_interaction" title="Resonant interaction">resonant interactions</a> in <a href="Dispersive_medium" class="mw-redirect" title="Dispersive medium">dispersive media</a>. They arise when three waves with <a href="Wave_vector" title="Wave vector">wave vectors</a> k1, k2, and k3 satisfy both the resonance condition (commonly expressed as k1 = k2 + k3) and the frequency matching condition ω1 = ω2 + ω3, where ωi denotes the <a href="Angular_frequency" title="Angular frequency">angular frequency</a> of each wave component. These resonant triad interactions enable efficient energy transfer between the three wave modes.

Because they provide a direct and tractable example of resonant wave interactions, have broad applicability across the physical sciences, and possess the remarkable property of <a href="Complete_integrability" class="mw-redirect" title="Complete integrability">complete integrability</a>, the three-wave equations have been extensively studied since the 1970s.<a href="#cite_note-zakharov-1">[1]</a> Their integrability allows for exact analytical solutions via methods such as the <a href="Inverse_scattering_transform" title="Inverse scattering transform">inverse scattering transform</a>, making them a cornerstone in the mathematical theory of <a href="Integrable_system" title="Integrable system">integrable systems</a> and in the study of <a href="Soliton" title="Soliton">soliton</a>-like phenomena. The equations have also played a crucial role in the development of <a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian</a> formulations of wave dynamics and in advancing the understanding of <a href="Energy_cascade" title="Energy cascade">energy cascades</a> in weakly nonlinear wave systems.

<div class="mw-heading mw-heading2"><h2 id="Informal_introduction">Informal introduction</h2></div>

The three-wave equation arises by consideration of some of the simplest imaginable <a href="Non-linear_system" class="mw-redirect" title="Non-linear system">non-linear systems</a>. Linear differential systems have the generic form

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<annotation encoding="application/x-tex">{\displaystyle D\psi =\lambda \psi }</annotation>

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</math><img src="./8fe2e744490441f4b51e89429203abf5689b11fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.404ex; height:2.509ex;" alt="{\displaystyle D\psi =\lambda \psi }" loading="lazy"></dd></dl>

for some <a href="Differential_operator" title="Differential operator">differential operator</a> D. The simplest non-linear extension of this is to write

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<annotation encoding="application/x-tex">{\displaystyle D\psi -\lambda \psi =\varepsilon \psi ^{2}.}</annotation>

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</math><img src="./1307a856f368c44de63ea57d61ba197dcc71bf24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.542ex; height:3.009ex;" alt="{\displaystyle D\psi -\lambda \psi =\varepsilon \psi ^{2}.}" loading="lazy"></dd></dl>

How can one solve this? Several approaches are available. In a few exceptional cases, there might be known exact solutions to equations of this form. In general, these are found in some ad hoc fashion after applying some <a href="Ansatz" title="Ansatz">ansatz</a>. A second approach is to assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon \ll 1}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon \ll 1}</annotation>

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</math><img src="./a9a9a39feb4c63deb7484f8f74a4bc7155d80793.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.86ex; height:2.176ex;" alt="{\displaystyle \varepsilon \ll 1}" loading="lazy"> and use <a href="Perturbation_theory" title="Perturbation theory">perturbation theory</a> to find "corrections" to the linearized theory. A third approach is to apply techniques from <a href="Scattering_matrix" class="mw-redirect" title="Scattering matrix">scattering matrix</a> (<a href="S-matrix" title="S-matrix">S-matrix</a>) theory.

In the S-matrix approach, one considers <a href="Particle" title="Particle">particles</a> or <a href="Plane_wave" title="Plane wave">plane waves</a> coming in from infinity, interacting, and then moving out to infinity. Counting from zero, the zero-particle case corresponds to the <a href="Vacuum" title="Vacuum">vacuum</a>, consisting entirely of the background. The one-particle case is a wave that comes in from the distant past and then disappears into thin air; this can happen when the background is absorbing, deadening or <a href="Dissipative" class="mw-redirect" title="Dissipative">dissipative</a>. Alternately, a wave appears out of thin air and moves away. This occurs when the background is unstable and generates waves: one says that the system "<a href="Radiation" title="Radiation">radiates</a>". The two-particle case consists of a particle coming in, and then going out. This is appropriate when the background is non-uniform: for example, an acoustic plane wave comes in, scatters from an enemy <a href="Submarine" title="Submarine">submarine</a>, and then moves out to infinity; by careful analysis of the outgoing wave, characteristics of the spatial inhomogeneity can be deduced. There are two more possibilities: <a href="Pair_creation" class="mw-redirect" title="Pair creation">pair creation</a> and <a href="Pair_annihilation" class="mw-redirect" title="Pair annihilation">pair annihilation</a>. In this case, a pair of waves is created "out of thin air" (by interacting with some background), or disappear into thin air.

Next on this count is the three-particle interaction. It is unique, in that it does not require any interacting background or vacuum, nor is it "boring" in the sense of a non-interacting plane-wave in a homogeneous background. Writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{1},\psi _{2},\psi _{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \psi _{1},\psi _{2},\psi _{3}}</annotation>

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</math><img src="./1964e613b7b73d7d9aa98f7a083aaf1ea2ffdc45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.77ex; height:2.509ex;" alt="{\displaystyle \psi _{1},\psi _{2},\psi _{3}}" loading="lazy"> for these three waves moving from/to infinity, this simplest quadratic interaction takes the form of

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<annotation encoding="application/x-tex">{\displaystyle (D-\lambda )\psi _{1}=\varepsilon \psi _{2}\psi _{3}}</annotation>

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</math><img src="./05277b62598f0569a6d4dc73f93b3b8b3b747d47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.813ex; height:2.843ex;" alt="{\displaystyle (D-\lambda )\psi _{1}=\varepsilon \psi _{2}\psi _{3}}" loading="lazy"></dd></dl>

and cyclic permutations thereof. This generic form can be called the three-wave equation; a specific form is presented below. A key point is that all quadratic <a href="Resonant_interaction" title="Resonant interaction">resonant interactions</a> can be written in this form (given appropriate assumptions). For time-varying systems where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> can be interpreted as <a href="Energy" title="Energy">energy</a>, one may write

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<annotation encoding="application/x-tex">{\displaystyle (D-i\partial /\partial t)\psi _{1}=\varepsilon \psi _{2}\psi _{3}}</annotation>

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</math><img src="./c7710971e981a0090dd5aadc2a0bf2b3810ed57f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.899ex; height:2.843ex;" alt="{\displaystyle (D-i\partial /\partial t)\psi _{1}=\varepsilon \psi _{2}\psi _{3}}" loading="lazy"></dd></dl>

for a time-dependent version.

<div class="mw-heading mw-heading2"><h2 id="Review">Review</h2></div>

Formally, the three-wave equation is

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial B_{j}}{\partial t}}+v_{j}\cdot \nabla B_{j}=\eta _{j}B_{\ell }^{*}B_{m}^{*}}</annotation>

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</math><img src="./6eb6aa13c760b241cb6f7d0508bf21c12fbe11ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.415ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial B_{j}}{\partial t}}+v_{j}\cdot \nabla B_{j}=\eta _{j}B_{\ell }^{*}B_{m}^{*}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j,\ell ,m=1,2,3}">

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<annotation encoding="application/x-tex">{\displaystyle j,\ell ,m=1,2,3}</annotation>

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</math><img src="./64b565d07d93578fbec7557a0689c126bc2ff918.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:14.716ex; height:2.509ex;" alt="{\displaystyle j,\ell ,m=1,2,3}" loading="lazy"> cyclic, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{j}}</annotation>

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</math><img src="./73fffa4919c0d6268f6a8d9f38c04dd3296fd0a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.037ex; height:2.343ex;" alt="{\displaystyle v_{j}}" loading="lazy"> is the <a href="Group_velocity" title="Group velocity">group velocity</a> for the wave having <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}_{j},\omega _{j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {k}}_{j},\omega _{j}}</annotation>

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</math><img src="./fc77693950e8a5202223fe786921482430972e5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.51ex; height:3.509ex;" alt="{\displaystyle {\vec {k}}_{j},\omega _{j}}" loading="lazy"> as the <a href="Wave-vector" class="mw-redirect" title="Wave-vector">wave-vector</a> and <a href="Angular_frequency" title="Angular frequency">angular frequency</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla }">

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<annotation encoding="application/x-tex">{\displaystyle \nabla }</annotation>

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</math><img src="./a3d0e93b78c50237f9ea83d027e4ebbdaef354b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }" loading="lazy"> the <a href="Gradient" title="Gradient">gradient</a>, taken in flat <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> in n dimensions. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{j}}">

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<annotation encoding="application/x-tex">{\displaystyle \eta _{j}}</annotation>

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</math><img src="./943ba755dfd551eb9ff2434aa5fae21a96bb1a58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.065ex; height:2.343ex;" alt="{\displaystyle \eta _{j}}" loading="lazy"> are the interaction coefficients; by rescaling the wave, they can be taken <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{j}=\pm 1}">

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<annotation encoding="application/x-tex">{\displaystyle \eta _{j}=\pm 1}</annotation>

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</math><img src="./fc0e538a94b3a87909fdf635147aab8c53ac60b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.134ex; height:2.843ex;" alt="{\displaystyle \eta _{j}=\pm 1}" loading="lazy">. By <a href="Cyclic_permutation" title="Cyclic permutation">cyclic permutation</a>, there are four classes of solutions. Writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =\eta _{1}\eta _{2}\eta _{3}}">

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<annotation encoding="application/x-tex">{\displaystyle \eta =\eta _{1}\eta _{2}\eta _{3}}</annotation>

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</math><img src="./19858312d1aa441ee32621e7948b528d52edb759.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.897ex; height:2.176ex;" alt="{\displaystyle \eta =\eta _{1}\eta _{2}\eta _{3}}" loading="lazy"> one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =\pm 1}">

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<annotation encoding="application/x-tex">{\displaystyle \eta =\pm 1}</annotation>

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</math><img src="./3a0fc62c3e2a5b3818c347dfb2f420801efac8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.238ex; height:2.676ex;" alt="{\displaystyle \eta =\pm 1}" loading="lazy">. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =-1}">

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<annotation encoding="application/x-tex">{\displaystyle \eta =-1}</annotation>

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</math><img src="./34de1550e84d142452c48896ea90983967972389.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.238ex; height:2.676ex;" alt="{\displaystyle \eta =-1}" loading="lazy"> are all equivalent under permutation. In 1+1 dimensions, there are three distinct <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =+1}">

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<annotation encoding="application/x-tex">{\displaystyle \eta =+1}</annotation>

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</math><img src="./52039fcc277a4b8e45f12016a4d5ae521b2b9d99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.238ex; height:2.676ex;" alt="{\displaystyle \eta =+1}" loading="lazy"> solutions: the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +++}">

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<annotation encoding="application/x-tex">{\displaystyle +++}</annotation>

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</math><img src="./f706537db2a916dd83197417e2bd1c8bb58ea0e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.457ex; height:2.176ex;" alt="{\displaystyle +++}" loading="lazy"> solutions, termed explosive; the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle --+}">

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<annotation encoding="application/x-tex">{\displaystyle --+}</annotation>

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</math><img src="./5c95968b90d82f7727e308ba15e2d27a1394bab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.457ex; height:2.176ex;" alt="{\displaystyle --+}" loading="lazy"> cases, termed <a href="Backscatter" title="Backscatter">stimulated backscatter</a>, and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -+-}">

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<annotation encoding="application/x-tex">{\displaystyle -+-}</annotation>

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</math><img src="./665744a3ee2c537362c335f0a71aa221b1f04197.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.457ex; height:2.176ex;" alt="{\displaystyle -+-}" loading="lazy"> case, termed <a href="Soliton" title="Soliton">soliton exchange</a>. These correspond to very distinct physical processes.<a href="#cite_note-2">[2]</a><a href="#cite_note-3">[3]</a> One interesting solution is termed the simulton, it consists of three comoving solitons, moving at a velocity v that differs from any of the three group velocities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},v_{2},v_{3}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{1},v_{2},v_{3}}</annotation>

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</math><img src="./12e5ac2b8e42965e3d97dbe06d1031361bc310e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.613ex; height:2.009ex;" alt="{\displaystyle v_{1},v_{2},v_{3}}" loading="lazy">. This solution has a possible relationship to the "three sisters" observed in <a href="Rogue_wave" title="Rogue wave">rogue waves</a>, even though deep water does not have a three-wave resonant interaction.

The lecture notes by Harvey Segur provide an introduction.<a href="#cite_note-segur-4">[4]</a>

The equations have a <a href="Lax_pair" title="Lax pair">Lax pair</a>, and are thus <a href="Completely_integrable" class="mw-redirect" title="Completely integrable">completely integrable</a>.<a href="#cite_note-zakharov-1">[1]</a><a href="#cite_note-5">[5]</a> The Lax pair is a 3x3 matrix pair, to which the <a href="Inverse_scattering_method" class="mw-redirect" title="Inverse scattering method">inverse scattering method</a> can be applied, using techniques by <a href="Athanassios_S._Fokas" class="mw-redirect" title="Athanassios S. Fokas">Fokas</a>.<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a> The class of spatially uniform solutions are known, these are given by <a href="Weierstrass's_elliptic_functions" class="mw-redirect" title="Weierstrass's elliptic functions">Weierstrass elliptic ℘-function</a>.<a href="#cite_note-martin-phd-8">[8]</a> The resonant interaction relations are in this case called the <a href="Manley%E2%80%93Rowe_relations" title="Manley–Rowe relations">Manley–Rowe relations</a>; the invariants that they describe are easily related to the <a href="J-invariant" title="J-invariant">modular invariants</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle g_{2}}</annotation>

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</math><img src="./f0261c34f2ad1e1b5317708b7f98ae13ee70ff1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.163ex; height:2.009ex;" alt="{\displaystyle g_{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{3}.}">

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<annotation encoding="application/x-tex">{\displaystyle g_{3}.}</annotation>

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</math><img src="./a60319e7be84634bb536f8c9460d50e094b553bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.81ex; height:2.009ex;" alt="{\displaystyle g_{3}.}" loading="lazy"><a href="#cite_note-martin-9">[9]</a>

That these appear is perhaps not entirely surprising, as there is a simple intuitive argument. Subtracting one wave-vector from the other two, one is left with two vectors that generate a <a href="Period_lattice" class="mw-redirect" title="Period lattice">period lattice</a>. All possible relative positions of two vectors are given by Klein's <a href="J-invariant" title="J-invariant">j-invariant</a>, thus one should expect solutions to be characterized by this.

A variety of exact solutions for various boundary conditions are known.<a href="#cite_note-10">[10]</a> A "nearly general solution" to the full non-linear PDE for the three-wave equation has recently been given. It is expressed in terms of five functions that can be freely chosen, and a <a href="Laurent_series" title="Laurent series">Laurent series</a> for the sixth parameter.<a href="#cite_note-martin-phd-8">[8]</a><a href="#cite_note-martin-9">[9]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Some selected applications of the three-wave equations include:

<ul><li>In <a href="Non-linear_optics" class="mw-redirect" title="Non-linear optics">non-linear optics</a>, <a href="Tunable_laser" title="Tunable laser">tunable lasers</a> covering a broad frequency spectrum can be created by <a href="Parametric_downconversion" class="mw-redirect" title="Parametric downconversion"> parametric three-wave mixing</a> in quadratic (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{(2)}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi ^{(2)}}</annotation>

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</math><img src="./fa36f7f6c02ec32b59cda49e12d59c098738017d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.789ex; height:3.176ex;" alt="{\displaystyle \chi ^{(2)}}" loading="lazy">) <a href="Nonlinear_crystal" class="mw-redirect" title="Nonlinear crystal">nonlinear crystals</a>.<a href="#cite_note-11">[11]</a></li>

<li><a href="Surface_acoustic_wave" title="Surface acoustic wave">Surface acoustic waves</a> and in electronic <a href="Parametric_amplifier" class="mw-redirect" title="Parametric amplifier">parametric amplifiers</a>.</li>

<li>Deep water waves do not in themselves have a three-wave interaction; however, this is evaded in multiple scenarios:

<ul><li>Deep-water <a href="Capillary_wave" title="Capillary wave">capillary waves</a> are described by the three-wave equation.<a href="#cite_note-segur-4">[4]</a></li>

<li>Acoustic waves couple to deep-water waves in a three-wave interaction,<a href="#cite_note-12">[12]</a></li>

<li><a href="Vortex" title="Vortex">Vorticity waves</a> couple in a triad.</li>

<li>A uniform current (necessarily spatially inhomogenous by depth) has triad interactions.</li></ul></li></ul>

<dl><dd>These cases are all naturally described by the three-wave equation.</dd></dl>

<ul><li>In <a href="Plasma_physics" class="mw-redirect" title="Plasma physics">plasma physics</a>, the three-wave equation describes coupling in plasmas.<a href="#cite_note-13">[13]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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