Inverse scattering transform

In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years. 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. In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years. 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. The inverse scattering transform was first introduced by Clifford S. Gardner, John M. Greene, and Martin D. Kruskal et al. (1967, 1974) 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 L 2 {displaystyle L^{2}} 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. 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 u t {displaystyle u_{t}} and u x {displaystyle u_{x}} denoting partial derivatives with respect to t and x, respectively. To solve the initial value problem for this equation where u ( x , 0 ) {displaystyle u(x,0)} is a known function of x, one associates to this equation the Schrödinger eigenvalue equation where ψ {displaystyle psi } is an unknown function of t and x and u is the solution of the Korteweg–de Vries equation that is unknown except at t = 0 {displaystyle t=0} . The constant λ {displaystyle lambda } is an eigenvalue.