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 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 formula_3 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 formula_4 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 formula_6 can be interpreted as energy, one may write for a time-dependent version. Review. Formally, the three-wave equation is where  formula_9 cyclic, formula_10 is the group velocity for the wave having formula_11 as the wave-vector and angular frequency, and formula_12 the gradient, taken in flat Euclidean space in "n" dimensions. The formula_13 are the interaction coefficients; by rescaling the wave, they can be taken formula_14. By cyclic permutation, there are four classes of solutions. Writing formula_15 one has formula_16. The formula_17 are all equivalent under permuation. In 1+1 dimensions, there are three distinct formula_18 solutions: the formula_19 solutions, termed "explosive"; the formula_20 cases, termed "stimulated backscatter", and the formula_21 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 formula_22. 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 formula_23 and formula_24 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: