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] The three-wave equation arises by consideration of some of the simplest imaginable non-linear systems.
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 third approach is to apply techniques from scattering matrix (S-matrix) theory.
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.
for these three waves moving from/to infinity, this simplest quadratic interaction takes the form of and cyclic permutations thereof.
A key point is that all quadratic resonant interactions can be written in this form (given appropriate assumptions).
can be interpreted as energy, one may write for a time-dependent version.
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.
[6][7] The class of spatially uniform solutions are known, these are given by Weierstrass elliptic ℘-function.
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] Some selected applications of the three-wave equations include: