Method for solving certain nonlinear partial differential equations
The 3-step algorithm: transform the initial solution to initial scattering data, evolve initial scattering data, transform evolved scattering data to evolved solution
In mathematics, the inverse scattering transform is a method that solves the initial value problem for a nonlinear partial differential equation using mathematical methods related to wave scattering.[1]: 4960 The direct scattering transform describes how a function scatters waves or generates bound-states.[2]: 39–43 The inverse scattering transform uses wave scattering data to construct the function responsible for wave scattering.[2]: 66–67 The direct and inverse scattering transforms are analogous to the direct and inverse Fourier transforms which are used to solve linear partial differential equations.[2]: 66–67
Using a pair of differential operators, a 3-step algorithm may solve nonlinear differential equations; the initial solution is transformed to scattering data (direct scattering transform), the scattering data evolves forward in time (time evolution), and the scattering data reconstructs the solution forward in time (inverse scattering transform).[2]: 66–67
This algorithm simplifies solving a nonlinear partial differential equation to solving 2 linear ordinary differential equations and an ordinary integral equation, a method ultimately leading to analytic solutions for many otherwise difficult to solve nonlinear partial differential equations.[2]: 72
The inverse scattering problem is equivalent to a Riemann–Hilbert factorization problem, at least in the case of equations of one space dimension.[3] This formulation can be generalized to differential operators of order greater than two and also to periodic problems.[4]
In higher space dimensions one has instead a "nonlocal" Riemann–Hilbert factorization problem (with convolution instead of multiplication) or a d-bar problem.
^Aktosun 2009.
^ abcdeDrazin & Johnson 1989.
^Ablowitz & Fokas 2003, pp. 604–620.
^Osborne 1995.
and 22 Related for: Inverse scattering transform information
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Martin Kruskal and Norman Zabusky in 1965, which led to the inversescatteringtransform method in 1967. In the special case of Hamiltonian systems, if...
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function over that line. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inversetransform. Radon further included...
A Fast Fourier Transform (FFT) is an algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis...
one class of nonlinear inverse problems was so before 1970, that of inverse spectral and (one space dimension) inversescattering problems, after the seminal...
overview of nonlinear water waves". Nonlinear Ocean Waves and the InverseScatteringTransform. Academic Press. pp. 3 ff. ISBN 978-0-12-528629-9. Alexander...
coordinate x and time t. An exact solution found by using the inversescatteringtransform is: u = 4 arctan ( 1 − ω 2 cos ( ω t ) ω cosh ( 1 − ω 2...
nonlinear evolution equation that may be solved by means of the inversescatteringtransform. It obeys an infinite number of conservation laws; it does not...
theory, an axiomatic basis for part of non-standard analysis Inversescatteringtransform, a method for solving some non-linear partial differential equations...
Kaup, David J.; Newell, Alan C.; Segur, Harvey (1974), "The inversescatteringtransform-Fourier analysis for nonlinear problems", Studies in Appl. Math...
obey the Pauli matrix algebra. Under periodic boundary conditions, the transformed Hamiltonian can be shown is of a very similar form: H ^ = − g J ∑ j =...
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