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.

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

[math]\displaystyle{ D\psi=\lambda\psi }[/math]

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

[math]\displaystyle{ 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]\displaystyle{ \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 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 [math]\displaystyle{ \psi_1, \psi_2, \psi_3 }[/math] for these three waves moving from/to infinity, this simplest quadratic interaction takes the form of

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

[math]\displaystyle{ (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]\displaystyle{ \frac{\partial B_j}{\partial t} + v_j \cdot \nabla B_j=\eta_j B^*_\ell B^*_m }[/math]

where [math]\displaystyle{ j,\ell,m=1,2,3 }[/math] cyclic, [math]\displaystyle{ v_j }[/math] is the group velocity for the wave having [math]\displaystyle{ \vec k_j, \omega_j }[/math] as the wave-vector and angular frequency, and [math]\displaystyle{ \nabla }[/math] the gradient, taken in flat Euclidean space in n dimensions. The [math]\displaystyle{ \eta_j }[/math] are the interaction coefficients; by rescaling the wave, they can be taken [math]\displaystyle{ \eta_j=\pm 1 }[/math]. By cyclic permutation, there are four classes of solutions. Writing [math]\displaystyle{ \eta=\eta_1\eta_2\eta_3 }[/math] one has [math]\displaystyle{ \eta=\pm 1 }[/math]. The [math]\displaystyle{ \eta=-1 }[/math] are all equivalent under permutation. In 1+1 dimensions, there are three distinct [math]\displaystyle{ \eta=+1 }[/math] solutions: the [math]\displaystyle{ +++ }[/math] solutions, termed explosive; the [math]\displaystyle{ --+ }[/math] cases, termed stimulated backscatter, and the [math]\displaystyle{ -+- }[/math] case, termed soliton exchange. These correspond to very distinct physical processes. 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]\displaystyle{ v_1, v_2, v_3 }[/math]. 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.

The equations have a Lax pair, and are thus completely integrable. The Lax pair is a 3x3 matrix pair, to which the inverse scattering method can be applied, using techniques by Fokas. The class of spatially uniform solutions are known, these are given by Weierstrass elliptic ℘-function. 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 [math]\displaystyle{ g_2 }[/math] and [math]\displaystyle{ g_3. }[/math] 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. 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.

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 ([math]\displaystyle{ \chi^{(2)} }[/math]) 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.
  • Acoustic waves couple to deep-water waves in a three-wave interaction,
  • 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.