- Source: 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
D
ψ
=
λ
ψ
{\displaystyle D\psi =\lambda \psi }
for some differential operator D. The simplest non-linear extension of this is to write
D
ψ
−
λ
ψ
=
ε
ψ
2
.
{\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
ε
≪
1
{\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
ψ
1
,
ψ
2
,
ψ
3
{\displaystyle \psi _{1},\psi _{2},\psi _{3}}
for these three waves moving from/to infinity, this simplest quadratic interaction takes the form of
(
D
−
λ
)
ψ
1
=
ε
ψ
2
ψ
3
{\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
λ
{\displaystyle \lambda }
can be interpreted as energy, one may write
(
D
−
i
∂
/
∂
t
)
ψ
1
=
ε
ψ
2
ψ
3
{\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
∂
B
j
∂
t
+
v
j
⋅
∇
B
j
=
η
j
B
ℓ
∗
B
m
∗
{\displaystyle {\frac {\partial B_{j}}{\partial t}}+v_{j}\cdot \nabla B_{j}=\eta _{j}B_{\ell }^{*}B_{m}^{*}}
where
j
,
ℓ
,
m
=
1
,
2
,
3
{\displaystyle j,\ell ,m=1,2,3}
cyclic,
v
j
{\displaystyle v_{j}}
is the group velocity for the wave having
k
→
j
,
ω
j
{\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
η
j
{\displaystyle \eta _{j}}
are the interaction coefficients; by rescaling the wave, they can be taken
η
j
=
±
1
{\displaystyle \eta _{j}=\pm 1}
. By cyclic permutation, there are four classes of solutions. Writing
η
=
η
1
η
2
η
3
{\displaystyle \eta =\eta _{1}\eta _{2}\eta _{3}}
one has
η
=
±
1
{\displaystyle \eta =\pm 1}
. The
η
=
−
1
{\displaystyle \eta =-1}
are all equivalent under permutation. In 1+1 dimensions, there are three distinct
η
=
+
1
{\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. 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
{\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.
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
g
2
{\displaystyle g_{2}}
and
g
3
.
{\displaystyle g_{3}.}
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 (
χ
(
2
)
{\displaystyle \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.
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.