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.

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 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 $L^{2}$ 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 :

{\frac {\partial u}{\partial t}}-6\,u\,{\frac {\partial u}{\partial x}}+{\frac {\partial ^{3}u}{\partial x^{3}}}=0,\,

with ${\frac {\partial u}{\partial t}}$ and ${\frac {\partial u}{\partial x}}$ denoting partial derivatives with respect to t and x.

To solve the initial value problem for this equation where $u(x,0)$ is a known function of x, one associates to this equation the Schrödinger eigenvalue equation

{\frac {\partial ^{2}\psi }{\partial x^{2}}}-u(x,t)\psi =\lambda \psi .

where $\psi$ is an unknown function of t and x and u is the solution of the Korteweg–de Vries equation that is unknown except at $t=0$ . The constant $\lambda$ is an eigenvalue.

From the Schrödinger equation we obtain

u={\frac {1}{\psi }}{\frac {\partial ^{2}\psi }{\partial x^{2}}}-\lambda .

Substituting this into the Korteweg–de Vries equation and integrating gives the equation

{\frac {\partial \psi }{\partial t}}+{\frac {\partial ^{3}\psi }{\partial x^{3}}}-3(u-\lambda ){\frac {\partial \psi }{\partial x}}=C\psi +D\psi \int {\frac {dx}{\psi ^{2}}}

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, $L$ and $M$ , such that $Lv=\lambda v$ and ${\frac {dv}{dt}}=Mv$ . It is extremely important that the eigenvalue $\lambda$ be independent of time; i.e. ${\frac {d\lambda }{dt}}=0.$ Necessary and sufficient conditions for this to occur are determined as follows: take the time derivative of $Lv=\lambda v$ to obtain

{\frac {dL}{dt}}v+L{\frac {dv}{dt}}={\frac {d\lambda }{dt}}v+\lambda {\frac {dv}{dt}}.

Plugging in $Mv$ for ${\frac {dv}{dt}}$ yields

{\frac {dL}{dt}}v+LMv={\frac {d\lambda }{dt}}v+\lambda Mv.

Rearranging on the far right term gives us

{\frac {dL}{dt}}v+LMv={\frac {d\lambda }{dt}}v+MLv.

Thus,

{\frac {dL}{dt}}v+LMv-MLv={\frac {d\lambda }{dt}}v.

Since $v\not =0$ , this implies that ${\frac {d\lambda }{dt}}=0$ if and only if

{\frac {dL}{dt}}+LM-ML=0.\,

This is Lax's equation. One important thing to note about Lax's equation is that ${\frac {dL}{dt}}$ 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):

L={\frac {d^{{2}}}{dx^{{2}}}}+u,

where u is the "potential". Comparing the expression ${\frac {dL}{dt}}v+L{\frac {dv}{dt}}$ with ${\frac {\partial }{\partial t}}\left({\frac {d^{{2}}v}{dx^{{2}}}}+uv\right)$ shows us that ${\frac {\partial L}{\partial t}}={\frac {du}{dt}},$ 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 $\lambda$ , 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;^[1] Vladimir Aleksandrovich Marchenko^[2]), 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. See Marchenko (1986) for a mathematical rigorous treatment.)

Examples of integrable equations

Further examples of integrable equations may be found on the article Integrable system.

References

↑ Gel’fand, I. M. & Levitan, B. M., "On the determination of a differential equation from its spectral function". American Mathematical Society Translations, (2)1:253–304, 1955.
↑ V. A. Marchenko, "Sturm-Liouville Operators and Applications", Birkhäuser, Basel, 1986.

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.
Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (1967), "Method for Solving the Korteweg-deVries Equation", Physical Review Letters, 19: 1095–1097, Bibcode:1967PhRvL..19.1095G, doi:10.1103/PhysRevLett.19.1095
Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. (1974), "Korteweg-deVries equation and generalization. VI. Methods for exact solution.", Comm. Pure Appl. Math., 27: 97–133, MR 0336122, doi:10.1002/cpa.3160270108
V. A. Marchenko, "Sturm-Liouville Operators and Applications", Birkhäuser, Basel, 1986.
J. Shaw, Mathematical Principles of Optical Fiber Communications, SIAM, Philadelphia, 2004.
Eds: R.K. Bullough, P.J. Caudrey. "Solitons" Topics in Current Physics 17. Springer Verlag, Berlin-Heidelberg-New York, 1980.

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