Inverse scattering transform

In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solve many linear partial differential equations. The name "inverse scattering method" comes from the key idea of recovering the time evolution of a potential from the time evolution of its scattering data: inverse scattering refers to the problem of recovering a potential from its scattering matrix, as opposed to the direct scattering problem of finding the scattering matrix from the potential.
The inverse scattering transform may be applied to many of the so-called exactly solvable models, that is to say completely integrable infinite dimensional systems. 
Overview.
The inverse scattering transform was first introduced by for the Korteweg–de Vries equation, and soon extended to the nonlinear Schrödinger equation, the Sine-Gordon equation, and the Toda lattice equation. It was later used to solve many other equations, such as the Kadomtsev–Petviashvili equation, the Ishimori equation, the Dym equation, and so on. A further family of examples is provided by the Bogomolny equations (for a given gauge group and oriented Riemannian 3-fold), the formula_1 solutions of which are magnetic monopoles.
A characteristic of solutions obtained by the inverse scattering method is the existence of solitons, solutions resembling both particles and waves, which have no analogue for linear partial differential equations. The term "soliton" arises from non-linear optics.
The inverse scattering problem can be written as a Riemann–Hilbert factorization problem, at least in the case of equations of one space dimension. This formulation can be generalized to differential operators of order greater than 2 and also to periodic potentials.
In higher space dimensions one has instead a "nonlocal" Riemann–Hilbert factorization problem (with convolution instead of multiplication) or a d-bar problem.
Example: the Korteweg–de Vries equation.
The Korteweg–de Vries equation is a nonlinear, dispersive, evolution partial differential equation for a function "u"; of two real variables, one space variable "x" and one time variable "t" :
with formula_3 and formula_4 denoting partial derivatives with respect to "t" and "x", respectively.
To solve the initial value problem for this equation where formula_5 is a known function of "x", one associates to this equation the Schrödinger eigenvalue equation
where formula_7 is an unknown function of "t" and "x" and "u" is the solution of the Korteweg–de Vries equation that is unknown except at formula_8. The constant formula_9 is an eigenvalue.
From the Schrödinger equation we obtain 
Substituting this into the Korteweg–de Vries equation and integrating gives the equation
where "C" and "D" are constants.
Method of solution.
Step 1. Determine the nonlinear partial differential equation. This is usually accomplished by analyzing the physics of the situation being studied.
Step 2. Employ "forward scattering". This consists in finding the Lax pair. The Lax pair consists of two linear operators, formula_12 and formula_13, such that formula_14 and formula_15. It is extremely important that the eigenvalue formula_9 be independent of time; i.e. formula_17 Necessary and sufficient conditions for this to occur are determined as follows: take the time derivative of formula_14 to obtain
Plugging in formula_20 for formula_21 yields
Rearranging on the far right term gives us
Thus,
Since formula_25, this implies that formula_26 if and only if
This is Lax's equation. In Lax's equation is that formula_28 is the time derivative of formula_12 precisely where it explicitly depends on formula_30. The reason for defining the differentiation this way is motivated by the simplest instance of formula_12, which is the Schrödinger operator (see Schrödinger equation):
where u is the "potential". Comparing the expression formula_33 with formula_34 shows us that formula_35 thus ignoring the first term.
After concocting the appropriate Lax pair it should be the case that Lax's equation recovers the original nonlinear PDE.
Step 3. Determine the time evolution of the eigenfunctions associated to each eigenvalue formula_9, the norming constants, and the reflection coefficient, all three comprising the so-called scattering data. This time evolution is given by a system of linear ordinary differential equations which can be solved.
Step 4. Perform the "inverse scattering" procedure by solving the Gelfand–Levitan–Marchenko integral equation (Israel Moiseevich Gelfand and Boris Moiseevich Levitan; Vladimir Aleksandrovich Marchenko), a linear integral equation, to obtain the final solution of the original nonlinear PDE. All the scattering data is required in order to do this. If the reflection coefficient is zero, the process becomes much easier. This step works if formula_12 is a differential or difference operator of order two, but not necessarily for higher orders. In all cases however, the "inverse scattering" problem is reducible to a Riemann–Hilbert factorization problem.
Examples of integrable equations.
Further examples of integrable equations may be found on the article Integrable system.