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
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
nonlinear
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
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.
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
:
for some
differential operator ''D''. The simplest non-linear extension of this is to write
:
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
and use
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...
to find "corrections" to the linearized theory. A third approach is to apply techniques from
scattering matrix (
S-matrix
In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
More forma ...
) theory.
In the S-matrix approach, one considers
particle
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, fro ...
s or
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, t ...
s coming in from infinity, interacting, and then moving out to infinity. Counting from zero, the zero-particle case corresponds to the
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
, 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
Radiata or Radiates is a historical taxonomic rank that was used to classify animals with radially symmetric body plans. The term Radiata is no longer accepted, as it united several different groupings of animals that do not form a monophyleti ...
". 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 particle physics, annihilation is the process that occurs when a subatomic particle collides with its respective antiparticle to produce other particles, such as an electron colliding with a positron to produce two photons. The total ene ...
. 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
for these three waves moving from/to infinity, this simplest quadratic interaction takes the form of
:
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
can be interpreted as
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
, one may write
:
for a time-dependent version.
Review
Formally, the three-wave equation is
:
where
cyclic,
is the
group velocity for the wave having
as the
wave-vector and
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
, and
the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
, taken in flat Euclidean space in ''n'' dimensions. The
are the interaction coefficients; by rescaling the wave, they can be taken
. By cyclic permutation, there are four classes of solutions. Writing
one has
. The
are all equivalent under permutation. In 1+1 dimensions, there are three distinct
solutions: the
solutions, termed ''explosive''; the
cases, termed ''
stimulated backscatter'', and the
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
. This solution has a possible relationship to the "three sisters" observed in
rogue wave
Rogue waves (also known as freak waves, monster waves, episodic waves, killer waves, extreme waves, and abnormal waves) are unusually large, unpredictable, and suddenly appearing surface waves that can be extremely dangerous to ships, even to lar ...
s, 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 In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss s ...
, and are thus
completely integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first i ...
.
[
] 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 and
[
]
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
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion ...
for the sixth parameter.
Applications
Some selected applications of the three-wave equations include:
* In
non-linear optics,
tunable laser
A tunable laser is a laser whose wavelength of operation can be altered in a controlled manner. While all laser gain media allow small shifts in output wavelength, only a few types of lasers allow continuous tuning over a significant wavelength ra ...
s covering a broad frequency spectrum can be created by
parametric three-wave mixing in quadratic (
)
nonlinear crystal Nonlinear photonic crystals are usually used as quasi-phase-matching materials. They can be one-dimensional, two-dimensional or three-dimensional.
Nonlinear Photonic Crystals
Broadly speaking, nonlinear photonic crystals (PC) are periodic structu ...
s.
*
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 wave
A capillary wave is a wave traveling along the phase boundary of a fluid, whose dynamics and phase velocity are dominated by the effects of surface tension.
Capillary waves are common in nature, and are often referred to as ripples. The ...
s 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
, the three-wave equation describes coupling in plasmas.
[
{{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
]
References
Nonlinear optics
Nonlinear systems
Differential equations