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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 equation]]s. 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.<ref name="zakharov"/>

==Informal introduction==
The three-wave equation arises by consideration of some of the simplest imaginable [[non-linear system]]s. Linear differential systems have the generic form 
:<math>D\psi=\lambda\psi</math>
for some [[differential operator]] ''D''.  The simplest non-linear extension of this is to write 
:<math>D\psi-\lambda\psi=\varepsilon\psi^2.</math>
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 [[ansatz]]. A second approach is to assume that <math>\varepsilon\ll 1</math> and use  [[perturbation theory]] to find "corrections" to the linearized theory. A third approach is to apply techniques from [[scattering matrix]] ([[S-matrix]]) theory.

In the S-matrix approach, one considers [[particle]]s or [[plane wave]]s coming in from infinity, interacting, and then moving out to infinity. Counting from zero, the zero-particle case corresponds to the [[vacuum]], 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 [[dissipative]]. 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 "[[radiation|radiates]]". 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 [[submarine]], 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: [[pair creation]] and [[pair annihilation]]. 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>\psi_1, \psi_2, \psi_3</math> for these three waves moving from/to infinity, this simplest quadratic interaction  takes the form of 
:<math>(D-\lambda)\psi_1=\varepsilon\psi_2\psi_3</math>
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 [[resonant interaction]]s can be written in this form (given appropriate assumptions). For time-varying systems where <math>\lambda</math> can be interpreted as [[energy]], one may write 
:<math>(D-i\partial/\partial t)\psi_1=\varepsilon\psi_2\psi_3</math>
for a time-dependent version.

==Review==
Formally, the three-wave equation is 
:<math>\frac{\partial B_j}{\partial t} + v_j \cdot \nabla B_j=\eta_j B^*_\ell B^*_m</math>
where <math>j,\ell,m=1,2,3</math> cyclic, <math>v_j</math> is the [[group velocity]] for the wave having <math>\vec k_j, \omega_j</math> as the [[wave-vector]] and [[angular frequency]], and <math>\nabla</math> the [[gradient]], taken in flat Euclidean space in ''n'' dimensions.  The <math>\eta_j</math> are the interaction coefficients; by rescaling the wave, they can be taken <math>\eta_j=\pm 1</math>. By cyclic permutation, there are four classes of solutions. Writing <math>\eta=\eta_1\eta_2\eta_3</math> one has <math>\eta=\pm 1</math>. The <math>\eta=-1</math> are all equivalent under permutation. In 1+1 dimensions, there are three distinct <math>\eta=+1</math> solutions: the <math>+++</math> solutions, termed ''explosive''; the <math>--+</math> cases, termed ''[[backscatter|stimulated backscatter]]'', and the <math>-+-</math> case, termed ''[[soliton|soliton exchange]]''. These correspond to very distinct physical processes.<ref>
{{cite journal
 |last1=Degasperis |first1=A.
 |last2=Conforti |first2=M.
 |last3=Baronio |first3=F.
 |last4=Wabnitz |first4=S.
 |last5=Lombardo |first5=S.
 |year=2011
 |title=The Three-Wave Resonant Interaction Equations: Spectral and Numerical Methods
 |url=https://hal.archives-ouvertes.fr/hal-02395151/file/3WNUMERFINALE.pdf
 |journal=[[Letters in Mathematical Physics]]
 |volume=96 |issue=1–3 |pages=367–403
 |bibcode=2011LMaPh..96..367D
 |doi=10.1007/s11005-010-0430-4
 |s2cid=18846092
}}</ref><ref>
{{cite journal
 |last1=Kaup |first1=D. J.
 |last2=Reiman |first2=A.
 |last3=Bers |first3=A.
 |year=1979
 |title=Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium
 |journal=[[Reviews of Modern Physics]]
 |volume=51 |issue=2 |pages=275–309
 |bibcode=1979RvMP...51..275K
 |doi=10.1103/RevModPhys.51.275
}}</ref> 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>v_1, v_2, v_3</math>. This solution has a possible relationship to the "three sisters" observed in [[rogue wave]]s, even though deep water does not have a three-wave resonant interaction.

The lecture notes by Harvey Segur provide an introduction.<ref name="segur">
{{cite web
 |last1=Segur |first1=H.
 |last2=Grisouard |first2=N.
 |year=2009
 |url=https://gfd.whoi.edu/wp-content/uploads/sites/18/2018/03/lecture13-harvey_136505.pdf
 |title=Lecture 13: Triad (or 3-wave) resonances
 |website=Geophysical Fluid Dynamics
 |publisher=[[Woods Hole Oceanographic Institution]]
}}</ref>

The equations have a [[Lax pair]], and are thus [[completely integrable]].<ref name="zakharov">
{{cite journal
 |last1=Zakharov |first1=V. E.
 |last2=Manakov |first2=S. V.
 |year=1975
 |title=On the theory of resonant interaction of wave packets in nonlinear media
 |journal=[[Soviet Physics JETP]]
 |volume=42 |issue=5 |pages=842–850
 |url=http://jetp.ac.ru/cgi-bin/dn/e_042_05_0842.pdf
}}</ref><ref>
{{cite book
 |last1=Zakharov |first1=V. E.
 |last2=Manakov |first2=S. V.
 |last3=Novikov |first3=S. P.
 |last4=Pitaevskii |first4=L. I.
 |year=1984
 |title=Theory of Solitons: The Inverse Scattering Method
 |publisher=[[Plenum Press]] |location=New York
 |bibcode=1984lcb..book.....N
}}</ref> The Lax pair is a 3x3 matrix pair, to which the [[inverse scattering method]] can be applied, using techniques by [[Athanassios S. Fokas|Fokas]].<ref>
{{cite journal
 |last1=Fokas |first1=A. S.
 |last2=Ablowitz |first2=M. J.
 |year=1984
 |title=On the inverse scattering transform of multidimensional nonlinear equations related to first‐order systems in the plane
 |journal=[[Journal of Mathematical Physics]]
 |volume=25 |issue=8 |pages=2494–2505
 |bibcode=1984JMP....25.2494F
 |doi=10.1063/1.526471
}}</ref><ref>
{{cite journal
 |last1=Lenells |first1=J.
 |year=2012
 |title=Initial-boundary value problems for integrable evolution equations with 3×3 Lax pairs
 |journal=[[Physica D]]
 |volume=241 |issue=8 |pages=857–875
 |arxiv=1108.2875
 |bibcode=2012PhyD..241..857L
 |doi=10.1016/j.physd.2012.01.010
 |s2cid=119144977
}}</ref> The class of spatially uniform solutions are known, these are given by [[Weierstrass's elliptic functions|Weierstrass elliptic ℘-function]].<ref name="martin-phd">
{{cite thesis
 |last1=Martin |first1=R. A.
 |year=2015
 |url=https://scholar.colorado.edu/concern/graduate_thesis_or_dissertations/7m01bk69s
 |title=Toward a General Solution of the Three-Wave Resonant Interaction Equations
 |publisher=[[University of Colorado]]
}}</ref> The resonant interaction relations are in this case called the [[Manley–Rowe relations]]; the invariants that they describe are easily related to the [[j-invariant|modular invariants]] <math>g_2</math> and <math>g_3.</math><ref name="martin">
{{cite journal
 |last1=Martin |first1=R. A.
 |last2=Segur |first2=H.
 |year=2016
 |title=Toward a General Solution of the Three-Wave Partial Differential Equations
 |journal=[[Studies in Applied Mathematics]]
 |volume=137 |pages=70–92
 |doi=10.1111/sapm.12133 |doi-access=free
}}</ref> 
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 [[period lattice]]. All possible relative positions of two vectors are given by Klein's [[j-invariant]], thus one should expect solutions to be characterized by this. 

A variety of exact solutions for various boundary conditions are known.<ref>
{{cite journal
 |last1=Kaup |first1=D. J.
 |year=1980
 |title=A Method for Solving the Separable Initial-Value Problem of the Full Three-Dimensional Three-Wave Interaction
 |journal=[[Studies in Applied Mathematics]]
 |volume=62 |pages=75–83
 |doi=10.1002/sapm198062175
}}</ref> 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 [[Laurent series]] for the sixth parameter.<ref name="martin-phd"/><ref name="martin"/>

==Applications==
Some selected applications of the three-wave equations include:
 
* In [[non-linear optics]], [[tunable laser]]s covering a broad frequency spectrum can be created by [[parametric downconversion| parametric three-wave mixing]] in quadratic (<math>\chi^{(2)}</math>) [[nonlinear crystal]]s.{{cn|date=September 2020}}
* [[Surface acoustic wave]]s and in electronic [[parametric amplifier]]s.
* Deep water waves do not in themselves have a three-wave interaction; however, this is evaded in multiple scenarios:
** Deep-water [[capillary wave]]s are described by the three-wave equation.<ref name="segur"/>
** Acoustic waves couple to deep-water waves in a three-wave interaction,<ref>
{{cite journal
 |last1=Kadri|first1=U.
 |year=2015
 |title=Triad Resonance in the Gravity–Acousic Family
 |journal=AGU Fall Meeting Abstracts
 |volume=2015
 |pages=OS11A–2006
 |bibcode=2015AGUFMOS11A2006K
 |doi=10.13140/RG.2.1.4283.1441 |doi-access=free
}}</ref> 
** [[vortex|Vorticity waves]] couple in a triad.
** A uniform current (necessarily spatially inhomogenous by depth) has triad interactions.
:These cases are all naturally described by the three-wave equation.
* In [[plasma physics]], the three-wave equation describes coupling in plasmas.<ref>
{{cite journal
 |last1=Kim|first1=J.-H.
 |last2=Terry|first2=P. W.
 |year=2011
 |title=A self-consistent three-wave coupling model with complex linear frequencies
 |url=https://zenodo.org/record/569793
 |journal=[[Physics of Plasmas]]
 |volume=18 |issue=9 |page=092308
 |bibcode=2011PhPl...18i2308K
 |doi=10.1063/1.3640807
}}</ref>

==References==
<references/>

[[Category:Nonlinear optics]]
[[Category:Nonlinear systems]]
[[Category:Differential equations]]