Three-wave equation

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.^[1]

Informal introduction

The three-wave equation arises by consideration of some of the simplest imaginable non-linear systems. Linear differential systems have the generic form

${\displaystyle D\psi =\lambda \psi }$ 

for some differential operator D. The simplest non-linear extension of this is to write

${\displaystyle D\psi -\lambda \psi =\varepsilon \psi ^{2}.}$ 

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 ${\displaystyle \varepsilon \ll 1}$  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 particles or plane waves 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 "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 ${\displaystyle \psi _{1},\psi _{2},\psi _{3}}$  for these three waves moving from/to infinity, this simplest quadratic interaction takes the form of

${\displaystyle (D-\lambda )\psi _{1}=\varepsilon \psi _{2}\psi _{3}}$ 

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 interactions can be written in this form (given appropriate assumptions). For time-varying systems where ${\displaystyle \lambda }$  can be interpreted as energy, one may write

${\displaystyle (D-i\partial /\partial t)\psi _{1}=\varepsilon \psi _{2}\psi _{3}}$ 

for a time-dependent version.

Review

Formally, the three-wave equation is

${\displaystyle {\frac {\partial B_{j}}{\partial t}}+v_{j}\cdot \nabla B_{j}=\eta _{j}B_{\ell }^{*}B_{m}^{*}}$ 

where ${\displaystyle j,\ell ,m=1,2,3}$  cyclic, ${\displaystyle v_{j}}$  is the group velocity for the wave having ${\displaystyle {\vec {k}}_{j},\omega _{j}}$  as the wave-vector and angular frequency, and ${\displaystyle \nabla }$  the gradient, taken in flat Euclidean space in n dimensions. The ${\displaystyle \eta _{j}}$  are the interaction coefficients; by rescaling the wave, they can be taken ${\displaystyle \eta _{j}=\pm 1}$ . By cyclic permutation, there are four classes of solutions. Writing ${\displaystyle \eta =\eta _{1}\eta _{2}\eta _{3}}$  one has ${\displaystyle \eta =\pm 1}$ . The ${\displaystyle \eta =-1}$  are all equivalent under permutation. In 1+1 dimensions, there are three distinct ${\displaystyle \eta =+1}$  solutions: the ${\displaystyle +++}$  solutions, termed explosive; the ${\displaystyle --+}$  cases, termed stimulated backscatter, and the ${\displaystyle -+-}$  case, termed soliton exchange. These correspond to very distinct physical processes.^[2][3] 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 ${\displaystyle v_{1},v_{2},v_{3}}$ . This solution has a possible relationship to the "three sisters" observed in rogue waves, even though deep water does not have a three-wave resonant interaction.

The lecture notes by Harvey Segur provide an introduction.^[4]

The equations have a Lax pair, and are thus completely integrable.^[1][5] The Lax pair is a 3x3 matrix pair, to which the inverse scattering method can be applied, using techniques by Fokas.^[6][7] The class of spatially uniform solutions are known, these are given by Weierstrass elliptic ℘-function.^[8] The resonant interaction relations are in this case called the Manley–Rowe relations; the invariants that they describe are easily related to the modular invariants ${\displaystyle g_{2}}$  and ${\displaystyle g_{3}.}$ ^[9] 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.^[10] 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.^[8][9]

Applications

Some selected applications of the three-wave equations include:

• In non-linear optics, tunable lasers covering a broad frequency spectrum can be created by parametric three-wave mixing in quadratic (${\displaystyle \chi ^{(2)}}$ ) nonlinear crystals.^{[citation needed]}
• Surface acoustic waves and in electronic parametric amplifiers.
• Deep water waves do not in themselves have a three-wave interaction; however, this is evaded in multiple scenarios:
  • Deep-water capillary waves are described by the three-wave equation.^[4]
  • Acoustic waves couple to deep-water waves in a three-wave interaction,^[11]
  • 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.^[12]

References

1. Zakharov, V. E.; Manakov, S. V. (1975). "On the theory of resonant interaction of wave packets in nonlinear media" (PDF). Soviet Physics JETP. 42 (5): 842–850.
2. Degasperis, A.; Conforti, M.; Baronio, F.; Wabnitz, S.; Lombardo, S. (2011). "The Three-Wave Resonant Interaction Equations: Spectral and Numerical Methods" (PDF). Letters in Mathematical Physics. 96 (1–3): 367–403. Bibcode:2011LMaPh..96..367D. doi:10.1007/s11005-010-0430-4. S2CID 18846092.
3. Kaup, D. J.; Reiman, A.; Bers, A. (1979). "Space-time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium". Reviews of Modern Physics. 51 (2): 275–309. Bibcode:1979RvMP...51..275K. doi:10.1103/RevModPhys.51.275.
4. Segur, H.; Grisouard, N. (2009). "Lecture 13: Triad (or 3-wave) resonances" (PDF). Geophysical Fluid Dynamics. Woods Hole Oceanographic Institution.
5. Zakharov, V. E.; Manakov, S. V.; Novikov, S. P.; Pitaevskii, L. I. (1984). Theory of Solitons: The Inverse Scattering Method. New York: Plenum Press. Bibcode:1984lcb..book.....N.
6. Fokas, A. S.; Ablowitz, M. J. (1984). "On the inverse scattering transform of multidimensional nonlinear equations related to first‐order systems in the plane". Journal of Mathematical Physics. 25 (8): 2494–2505. Bibcode:1984JMP....25.2494F. doi:10.1063/1.526471.
7. Lenells, J. (2012). "Initial-boundary value problems for integrable evolution equations with 3×3 Lax pairs". Physica D. 241 (8): 857–875. arXiv:1108.2875. Bibcode:2012PhyD..241..857L. doi:10.1016/j.physd.2012.01.010. S2CID 119144977.
8. Martin, R. A. (2015). Toward a General Solution of the Three-Wave Resonant Interaction Equations (Thesis). University of Colorado.
9. Martin, R. A.; Segur, H. (2016). "Toward a General Solution of the Three-Wave Partial Differential Equations". Studies in Applied Mathematics. 137: 70–92. doi:10.1111/sapm.12133.
10. Kaup, D. J. (1980). "A Method for Solving the Separable Initial-Value Problem of the Full Three-Dimensional Three-Wave Interaction". Studies in Applied Mathematics. 62: 75–83. doi:10.1002/sapm198062175.
11. Kadri, U. (2015). "Triad Resonance in the Gravity–Acousic Family". AGU Fall Meeting Abstracts. 2015: OS11A–2006. Bibcode:2015AGUFMOS11A2006K. doi:10.13140/RG.2.1.4283.1441.
12. Kim, J.-H.; Terry, P. W. (2011). "A self-consistent three-wave coupling model with complex linear frequencies". Physics of Plasmas. 18 (9): 092308. Bibcode:2011PhPl...18i2308K. doi:10.1063/1.3640807.