Three-wave equation explained

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.

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

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

\psi _{1,} \psi _{2,} \psi _{3} for these three waves moving from/to infinity, this simplest quadratic interaction takes the form of (D -λ )\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 λ can be interpreted as energy, one may write (D - i\partial/\partial t)\psi _{1=\varepsilon\psi} _{2\psi} _{3} for a time-dependent version.

Review

Formally, the three-wave equation is

(\partial B _{j})/(\partial t) + v _{j} ⋅ \nabla B _{j=η} _{j} B ^{*} _{\ell} B ^{*} _{m} where j,\ell,m =1 , 2 , 3 cyclic, v _{j} is the group velocity for the wave having \vec k _{j,} \omega _{j} as the wave-vector and angular frequency, and \nabla the gradient, taken in flat Euclidean space in n dimensions. The η _{j} are the interaction coefficients; by rescaling the wave, they can be taken η _{j=\pm} 1 . By cyclic permutation, there are four classes of solutions. Writing η=η _{1η} _{2η} _{3} one has η= \pm 1 . The η=-1 are all equivalent under permutation. In 1+1 dimensions, there are three distinct η=+1 solutions: the +++ solutions, termed explosive; the --+ cases, termed stimulated backscatter, and the -+- case, termed soliton exchange. These correspond to very distinct physical processes.^{[1] [2]} 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 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.^[3]

g _{2} and 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 (

\chi ^{(2)} ) nonlinear crystals.

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