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In mathematics, the inverse scattering transform is a procedure for integrating certain nonlinear partial differential equations (PDEs) by first converting them into a system of linear ordinary differential equations (ODEs). The basic idea is not unlike the Fourier transform. It applies to "potentials" (see below for an example) that are rapidly decaying at infinity. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ...
In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. ...
In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...
The inverse scattering transform may be applied to many of the so-called exactly solvable models. These include the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schödinger equations, the Sine-Gordon equation, and the Dym equation. Solutions typically consist of solitons plus some background radiation decaying to zero as time goes to infinity, and are characterized by having non-obvious and un-intuitive constants of motion. In theoretical physics, an exactly solvable model or integrable model refers to a physical model, a physical theory, or set of differential equations whose exact solution may be calculated analytically in terms of elementary or special functions; the adjective integrable is therefore implies solvablility. ...
The Korteweg-de Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t: Its solutions clump up into solitons. ...
In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödingers equation in two dimensions. ...
The Sine-Gordon equation is a partial differential equation for a function of two real variables, x and t, given as follows: The name is a pun on the Klein-Gordon equation. ...
In mathematics, and in particular in the theory of solitons, the Dym equation (also known as DH) is named for Harry Dym. ...
In mathematics and physics, a soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ...
In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. ...
The inverse scattering problem can be written as a Riemann-Hilbert factorization problem. Such a "modern" formulation can be generalized to differential operators of order greater than two, but also to the case of periodic potentials.
Method of solution
Step 1. Determine the nonlinear partial differential equation. This is usually accomplished by analyzing the physics of the situation being studied. Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the branch of science concerned with the fundamental laws of the universe. ...
Step 2. Employ forward scattering. This consists of finding the Lax pair. The Lax pair consists of two linear operators, L and M, such that Lv = λv and vt = Mv. It is extremely important that the eigenvalue λ be independent of time; i.e. λt = 0. Necessary and sufficient conditions for this to occur are determined as follows: take the time derivative of Lv = λv to obtain In mathematics, in the theory of differential equations, a Lax pair is a pair of time-dependent matrices that describe certain solutions of differential equations. ...
In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...
In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...
In some places this article assumes an acquaintance with algebra, analytic geometry, or the limit. ...
- Ltv + Lvt = λtv + λvt.
Plugging in Mv for vt yields - Ltv + LMv = λtv + λMv.
Rearranging on the far right term gives us - Ltv + LMv = λtv + MLv.
Thus, - Ltv + LMv − MLv = λtv.
Since , this implies that λt = 0 if and only if IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...
- Lt + LM − ML = 0.
This is Lax's equation. One important thing to note about Lax's equation is that Lt is the time derivative of L precisely where it explicitly depends on t. The reason for defining the differentiation this way is motivated by the simplest instance of L, which is the Schrödinger operator (see Schrödinger equation): In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the space- and time-dependence of quantum mechanical systems. ...
where u is the "potential". Comparing the expression Ltv + Lvt with shows us that 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 eigenvalues λ, the norming constants, and the reflection coefficient, all three comprising the so-called scattering data. This is all a linear process, though complicated.
Step 4. Perform the inverse scattering procedure by solving the Marchenko equation, 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. Note that if the reflection coefficient is zero, the process becomes much easier. Note also that this step works if L 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. (See Ablowitz-Clarkson (1991) for either approach.) In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ...
References - M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.
- N. Asano, Y. Kato, Algebraic and Spectral Methods for Nonlinear Wave Equations, Longman Scientific & Technical, Essex, England, 1990.
- M. Ablowitz, P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.
- J. Shaw, Mathematical Principles of Optical Fiber Communications, SIAM, Philadelphia, 2004.
External Links - Introductory paper on ISTPDF (300 KiB)
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