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{{Short description|Method for solving certain nonlinear partial differential equations}}

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

[[File:Inverse scattering transform.png|thumb|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 system|nonlinear]] [[partial differential equation]] using mathematical methods related to [[scattering|wave scattering]].{{sfn|Aktosun|2009}}{{rp|4960}} The <em>direct scattering transform</em> describes how a [[Function (mathematics)|function]] scatters waves or generates [[Bound state|bound-states]].{{sfn|Drazin|Johnson|1989}}{{rp|39-43}} The <em>inverse scattering transform</em> uses wave scattering data to construct the function responsible for wave scattering.{{sfn|Drazin|Johnson|1989}}{{rp|66-67}} The direct and inverse scattering transforms are analogous to the direct and inverse [[Fourier transform]]s which are used to solve [[Linear differential equation|linear]] partial differential equations.{{sfn|Drazin|Johnson|1989}}{{rp|66-67}}

Using a pair of [[differential operator]]s, a 3-step algorithm may solve [[nonlinear system|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).{{sfn|Drazin|Johnson|1989}}{{rp|66-67}}

The inverse scattering transform may be applied to many of the so-called [[exactly solvable model]]s, that is to say completely integrable infinite dimensional systems. It was first introduced by {{harvs|txt|mr=0336122|last1=Gardner|first1=Clifford S.|last2= Greene|first2= John M.|last3= Kruskal|first3= Martin D.|last4= Miura|first4= Robert M.|year=1974|pages= 97–133}} 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 [[Bogomol'nyi–Prasad–Sommerfield bound|Bogomolny equations]] (for a given gauge group and oriented Riemannian 3-fold), the <math>L^2</math> solutions of which are [[magnetic monopoles]].

This algorithm simplifies solving a nonlinear partial differential equation to solving 2 linear [[ordinary differential equation]]s and an ordinary [[integral equation]], a method ultimately leading to [[Analytic function|analytic solutions]] for many otherwise difficult to solve nonlinear partial differential equations.{{sfn|Drazin|Johnson|1989}}{{rp|72}}

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 is equivalent to a [[Riemann–Hilbert factorization]] problem, at least in the case of equations of one space dimension.{{sfn|Ablowitz|Fokas|2003|pp=604-620}} This formulation can be generalized to differential operators of order greater than two and also to periodic problems.{{sfn|Osborne|1995}}

In higher space dimensions one has instead a "nonlocal" Riemann–Hilbert factorization problem (with convolution instead of multiplication) or a d-bar problem.

==History==

The inverse scattering problem can be written as a [[Riemann–Hilbert factorization]] problem. This formulation can be generalized to differential operators of order greater than 2 and also to periodic potentials.

The inverse scattering transform arose from studying solitary waves. [[John Scott Russell|J.S. Russell]] described a "wave of translation" or "solitary wave" occurring in shallow water.{{sfn|Ablowitz|2023}} First [[Joseph Valentin Boussinesq|J.V. Boussinesq]] and later [[Diederik Korteweg|D. Korteweg]] and [[Gustav de Vries|G. deVries]] discovered the [[Korteweg–de Vries equation|Korteweg-deVries (KdV) equation]], a nonlinear partial differential equation describing these waves.{{sfn|Ablowitz|2023}} Later, N. Zabusky and M. Kruskal, using numerical methods for investigating the [[Fermi–Pasta–Ulam–Tsingou problem]], found that solitary waves had the elastic properties of colliding particles; the waves' initial and ultimate amplitudes and velocities remained unchanged after wave collisions.{{sfn|Ablowitz|2023}} These particle-like waves are called [[soliton]]s and arise in nonlinear equations because of a weak balance between dispersive and nonlinear effects.{{sfn|Ablowitz|2023}}

Gardner, Greene, Kruskal and Miura introduced the inverse scattering transform for solving the [[Korteweg–de Vries equation]].{{sfn|Gardner|Greene|Kruskal|Miura|1967}} Lax, Ablowitz, Kaup, Newell, and Segur generalized this approach which led to solving other nonlinear equations including the [[nonlinear Schrödinger equation]], [[sine-Gordon equation]], [[Korteweg–De Vries equation|modified Korteweg–De Vries equation]], [[Kadomtsev–Petviashvili equation]], the [[Ishimori equation]], [[Toda lattice]] equation, and the [[Dym equation]].{{sfn|Ablowitz|2023}}{{sfn|Konopelchenko|Dubrowsky|1991}}{{sfn|Oono|1996}} This approach has also been applied to different types of nonlinear equations including differential-difference, partial difference, multidimensional equations and fractional integrable nonlinear systems.{{sfn|Ablowitz|2023}}

==Example: the Korteweg–de Vries equation==

==Description==

The Korteweg–de Vries equation is a nonlinear, dispersive, evolution [[partial differential equation]] for a [[function (mathematics)|function]] ''u''; of two [[real number|real]] variables, one space variable ''x'' and one time variable ''t'' :

===Nonlinear partial differential equation===

The independent variables are a spatial variable <math>x</math> and a time variable <math>t</math>. Subscripts or differential operators (<math display="inline"> \partial_{x}, \partial_{t} </math>) indicate differentiation. The function <math>u(x,t)</math> is a solution of a nonlinear partial differential equation, <math display="inline">u_{t}+N(u)=0</math>, with [[initial condition|initial condition (value)]] <math display="inline">u(x,0)</math>.{{sfn|Drazin|Johnson|1989}}{{rp|72}}

===Requirements===

:<math> \frac{\partial u}{\partial t}- 6\, u\,

The differential equation's solution meets the integrability and Fadeev conditions:{{sfn|Drazin|Johnson|1989}}{{rp|40}}

\frac{\partial u}{\partial x}+

:Integrability condition:<math>\int^{\infty}_{-\infty} \ |u(x)| \ dx \ < \infty</math>

\frac{\partial^3 u}{\partial x^3} =0,\,</math>

:Fadeev condition: <math>\int^{\infty}_{-\infty} \ (1+|x|)|u(x)| \ dx \ < \infty</math>

===Differential operator pair===

with <math> \frac{\partial u}{\partial t}</math> and <math> \frac{\partial u}{\partial x}</math> denoting [[partial derivative]]s with respect to ''t'' and ''x''.

The [[Lax pair|Lax differential operators]], <math display="inline">L</math> and <math display="inline">M</math>, are linear ordinary differential operators with coefficients that may contain the function <math display="inline">u(x,t)</math> or its derivatives. The [[self-adjoint operator]] <math display="inline">L</math> has a time derivative <math display="inline">L_{t}</math> and generates a <em>eigenvalue (spectral) equation</em> with [[eigenfunction]]s <math display="inline">\psi</math> and time-constant [[eigenvalues and eigenvectors|eigenvalues]] (<em>[[Spectral theory|spectral parameters]]</em>) <math display="inline">\lambda</math>.{{sfn|Aktosun|2009}}{{rp|4963}}{{sfn|Drazin|Johnson|1989}}{{rp|98}}

: <math> L(\psi)=\lambda \psi , \ </math> and <math display="inline"> \ L_{t}(\psi) \overset{def}{=}(L(\psi))_{t}-L(\psi_{t})</math>

The operator <math display="inline">M</math> describes how the eigenfunctions evolve over time, and generates a new eigenfunction <math display="inline">\widetilde{\psi} </math> of operator <math display="inline">L</math> from eigenfunction <math display="inline">\psi</math> of <math display="inline">L</math>.{{sfn|Aktosun|2009}}{{rp|4963}}

: <math>\widetilde{\psi}=\psi_{t}-M(\psi) \ </math>

The Lax operators combine to form a multiplicative operator, not a differential operator, of the eigenfunctions <math display="inline">\psi</math>.{{sfn|Aktosun|2009}}{{rp|4963}}

: <math>(L_{t}+LM-ML)\psi=0</math>

The Lax operators are chosen to make the multiplicative operator equal to the nonlinear differential equation.{{sfn|Aktosun|2009}}{{rp|4963}}

: <math>L_{t}+LM-ML=u_{t}+N(u)=0</math>

The [[AKNS system|AKNS differential operators]], developed by Ablowitz, Kaup, Newell, and Segur, are an alternative to the Lax differential operators and achieve a similar result.{{sfn|Aktosun|2009}}{{rp|4964}}{{sfn|Ablowitz|Kaup|Newell|Segur|1973}}{{sfn|Ablowitz|Kaup|Newell|Segur|1974}}

===Direct scattering transform===

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

The direct scattering transform generates initial scattering data; this may include the reflection coefficients, transmission coefficient, eigenvalue data, and normalization constants of the eigenfunction solutions for this differential equation.{{sfn|Drazin|Johnson|1989}}{{rp|39-48}}

: <math> L(\psi)=\lambda \psi </math>

===Scattering data time evolution===

The equations describing how scattering data evolves over time occur as solutions to a 1st order linear ordinary differential equation with respect to time. Using varying approaches, this first order linear differential equation may arise from the linear differential operators (Lax pair, AKNS pair), a combination of the linear differential operators and the nonlinear differential equation, or through additional substitution, integration or differentiation operations. Spatially asymptotic equations (<math display="inline">x \to \pm \infty</math>) simplify solving these differential equations.{{sfn|Aktosun|2009}}{{rp|4967-4968}}{{sfn|Drazin|Johnson|1989}}{{rp|68-72}}{{sfn|Gardner|Greene|Kruskal|Miura|1967}}

===Inverse scattering transform===

:<math> \frac{\partial^2 \psi}{\partial x^2}-u(x,t)\psi=\lambda\psi.</math>

The [[Marchenko equation|Marchenko]] equation combines the scattering data into a linear [[Fredholm integral equation]]. The solution to this integral equation leads to the solution, u(x,t), of the nonlinear differential equation.{{sfn|Drazin|Johnson|1989}}{{rp|48-57}}

==Example: Korteweg–De Vries equation==

where <math>\psi</math> is an unknown function of ''t'' and ''x'' and ''u'' is the solution of the Korteweg–de Vries equation that is unknown except at <math>t=0</math>. The constant <math>\lambda</math> is an eigenvalue.

The nonlinear differential Korteweg–De Vries equation is

{{sfn|Ablowitz|Segur|1981}}{{rp|4}}

: <math>u_{t}-6uu_{x}+u_{xxx}=0</math>

===Lax operators===

The Lax operators are:{{sfn|Drazin|Johnson|1989}}{{rp|97-102}}

: <math>L= -\partial^{2}_{x}+u(x,t) \ </math> and <math display="inline"> \ M= -4\partial^{3}_{x}+6u\partial_{x}+3u_{x} </math>

The multiplicative operator is:

: <math>L_{t}+LM-ML=u_{t}-6uu_{x}+u_{xxx}=0</math>

===Direct scattering transform===

From the Schrödinger equation we obtain

The solutions to this differential equation

:<math> u=\frac{1}{\psi} \frac{\partial^2 \psi}{\partial x^2} - \lambda.</math>

: <math display="inline">L(\psi)=-\psi_{xx}+u(x,0)\psi= \lambda \psi</math>

may include <em>scattering solutions</em> with a continuous range of eigenvalues (<em>continuous spectrum</em>) and <em>[[bound state|bound-state]]</em> solutions with discrete eigenvalues (<em>discrete spectrum</em>). The scattering data includes transmission coefficients <math display="inline">T(k,0)</math>, left reflection coefficient <math display="inline">R_{L}(k,0)</math>, right reflection coefficient <math display="inline">R_{R}(k,0)</math>, discrete eigenvalues <math display="inline">-\kappa^{2}_{1}, \ldots,-\kappa^{2}_{N}</math>, and left and right bound-state <em>normalization (norming) constants</em>.{{sfn|Aktosun|2009}}{{rp|4960}}

: <math>c(0)_{Lj}=\left( \int^{\infty}_{-\infty} \ \psi^{2}_{L}(ik_{j},x,0) \ dx \right)^{-1/2} \ j=1, \dots, N </math>

: <math> c(0)_{Rj}=\left( \int^{\infty}_{-\infty} \ \psi^{2}_{R}(ik_{j},x,0) \ dx \right)^{-1/2} \ j=1, \dots, N </math>

===Scattering data time evolution===

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

The spatially asymptotic left <math display="inline">\psi_{L}(k,x,t)</math> and right <math display="inline">\psi_{R}(k,x,t)</math> [[Jost function]]s simplify this step.{{sfn|Aktosun|2009}}{{rp|4965-4966}}

:<math> \begin{align}

\psi_{L}(x,k,t)&=e^{ikx}+o(1), \ x \to +\infty \\

\psi_{L}(x,k,t)&=\frac{e^{ikx}}{T(k,t)}+\frac{R_{L}(k,t)e^{-ikx}}{T(k,t)}+o(1), \ x \to - \infty \\

\psi_{R}(x,k,t)&=\frac{e^{-ikx}}{T(k,t)}+\frac{R_{R}(k,t)e^{ikx}}{T(k,t)}+o(1), \ x \to +\infty \\

\psi_{R}(x,k,t)&=e^{-ikx}+o(1), \ x \to -\infty \\

\end{align}</math>

The <em>dependency constants</em> <math display="inline"> \gamma_{j}(t)</math> relate the right and left Jost functions and right and left normalization constants.{{sfn|Aktosun|2009}}{{rp|4965-4966}}

:<math>\gamma_{j}(t)=\frac{\psi_{L}(x,i\kappa_{j},t)}{\psi_{R}(x,i\kappa_{j},t)}=(-1)^{N-j} \frac{c_{Rj}(t)}{c_{Lj}(t)}</math>

The Lax <math display="inline">M</math> differential operator generates an eigenfunction which can be expressed as a time-dependent linear combination of other eigenfunctions.{{sfn|Aktosun|2009}}{{rp|4967}}

:<math>\partial_{t}\psi_{L}(k,x,t)-M\psi_{L}(x,k,t)=

a_{L}(k,t)\psi_{L}(x,k,t)+b_{L}(k,t)\psi_{R}(x,k,t) </math>

:<math>\partial_{t}\psi_{R}(k,x,t)-M\psi_{R}(x,k,t)=

a_{R}(k,t)\psi_{L}(x,k,t)+b_{R}(k,t)\psi_{R}(x,k,t) </math>

The solutions to these differential equations, determined using scattering and bound-state spatially asymptotic Jost functions, indicate a time-constant transmission coefficient <math display="inline">T(k,t)</math>, but time-dependent reflection coefficients and normalization coefficients.{{sfn|Aktosun|2009}}{{rp|4967-4968}}

: <math>\begin{align}

R_{L}(k,t)&=R_{L}(k,0)e^{-i8k^{3}t} \\

R_{R}(k,t)&=R_{R}(k,0)e^{+i8k^{3}t} \\

c_{Lj}(t)&=c_{Lj}(0)e^{+4\kappa^{3}_{j}t}, \ j=1, \ldots, N \\

c_{Rj}(t)&=c_{Rj}(0)e^{-4\kappa^{3}_{j}t}, \ j=1, \ldots, N \end{align}</math>

===Inverse scattering transform===

:<math> \frac{\partial \psi}{\partial t}+\frac{\partial^3 \psi}{\partial x^3}-3(u-\lambda)

The <em>Marchenko kernel</em> is <math display="inline">F(x,t)</math>.{{sfn|Aktosun|2009}}{{rp|4968-4969}}

\frac{\partial \psi}{\partial x}=C\psi+D\psi\int \frac{dx}{\psi^2}</math>

:<math>F(x,t)\overset{def}{=}\frac{1}{2 \pi} \int^{\infty}_{-\infty}

R_{R}(k,t)

e^{ikx} \ dk + \sum^{N}_{j=1} c(t)^{2}_{Lj}e^{-\kappa_{j}x} </math>

The [[Gelfand–Levitan–Marchenko integral equation|Marchenko integral equation]] is a linear integral equation solved for <math display="inline">K(x,y,t)</math>.{{sfn|Aktosun|2009}}{{rp|4968-4969}}

where ''C'' and ''D'' are constants.

: <math> K(x,z,t)+F(x+z,t)+ \int^{\infty}_{x} K(x,y,t)F(y+z,t) \ dy=0 </math>

The solution to the Marchenko equation, <math display="inline">K(x,y,t) </math>, generates the solution <math display="inline">u(x,t)</math> to the nonlinear partial differential equation.{{sfn|Aktosun|2009}}{{rp|4969}}

==Method of solution==

:<math>u(x,t)= -2 \frac{\partial K(x,x,t)}{\partial x}</math>

'''Step 1.''' Determine the nonlinear partial differential equation. This is usually accomplished by analyzing the [[physics]] of the situation being studied.

==Examples of integrable equations==

'''Step 2.''' Employ ''forward scattering''. This consists in finding the [[Lax pair]]. The Lax pair consists of two linear [[Operator (mathematics)|operator]]s, <math>L</math> and <math>M</math>, such that <math>Lv=\lambda v</math> and <math>\frac{dv}{dt}=Mv</math>. It is extremely important that the [[eigenvalue]] <math>\lambda</math> be independent of time; i.e. <math>\frac{d\lambda}{dt}=0.</math> Necessary and sufficient conditions for this to occur are determined as follows: take the time [[derivative]] of <math>Lv=\lambda v</math> to obtain

* [[Korteweg–de Vries equation]]

* [[nonlinear Schrödinger equation]]

* [[Camassa-Holm equation]]

* [[Sine-Gordon equation]]

* [[Toda lattice]]

* [[Ishimori equation]]

* [[Dym equation]]

== See also ==

:<math>\frac{dL}{dt}v+L\frac{dv}{dt}=\frac{d\lambda}{dt}v+\lambda \frac{dv}{dt}.</math>

* [[Quantum inverse scattering method]]

* [[Integrable system#List of some well-known classical integrable systems|Integrable system]]

==Citations==

Plugging in <math>Mv</math> for <math>\frac{dv}{dt}</math> yields

{{reflist}}

==References==

:<math>\frac{dL}{dt}v+LMv=\frac{d\lambda}{dt}v+\lambda Mv.</math>

*{{cite journal |last1=Ablowitz |first1=M. J. |last2=Kaup |first2=D. J. |last3=Newell |first3=A. C. |last4=Segur |first4=H. |title=Method for Solving the Sine-Gordon Equation |journal=Physical Review Letters |year=1973 |volume=30 |issue=25 |pages=1262–1264 |doi=10.1103/PhysRevLett.30.1262 |bibcode=1973PhRvL..30.1262A |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.30.1262|url-access=subscription }}

* {{cite journal |last1=Ablowitz |first1=M.J. |last2=Kaup |first2=D.J. |last3=Newell |first3=A.C. |last4=Segur |first4=H. |title=The Inverse Scattering Transform—Fourier Analysis for Nonlinear Problems |journal=Studies in Applied Mathematics |year=1974 |volume=53 |issue=4 |pages=249–315 |doi=10.1002/sapm1974534249 |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/sapm1974534249|url-access=subscription }}

Rearranging on the far right term gives us

*{{cite book |last1=Ablowitz |first1=Mark J. |last2=Segur |first2=Harvey |title=Solitons and the Inverse Scattering Transform |year=1981 |publisher=SIAM |isbn=978-0-89871-477-7 |url=https://books.google.com/books?id=Bzu4XAUpFZUC |language=en}}

:<math>\frac{dL}{dt}v+LMv=\frac{d\lambda}{dt}v+MLv.</math>

*{{cite book |last1=Ablowitz |first1=Mark J. |last2=Fokas |first2=A. S. |title=Complex Variables: Introduction and Applications |year=2003 |publisher=Cambridge University Press |isbn=978-0-521-53429-1 |pages=604–620 |url=https://books.google.com/books?id=SFqbV3i3hO0C |language=en}}

Thus,

*{{cite journal |last1=Ablowitz |first1=Mark J. |title=Nonlinear waves and the Inverse Scattering Transform |journal=Optik |year=2023 |volume=278 |pages=170710 |doi=10.1016/j.ijleo.2023.170710 |bibcode=2023Optik.27870710A |url=https://www.sciencedirect.com/science/article/pii/S0030402623002061|url-access=subscription }}

:<math>\frac{dL}{dt}v+LMv-MLv=\frac{d\lambda}{dt}v.</math>

*{{cite book |last1=Aktosun |first1=Tuncay |title=Encyclopedia of Complexity and Systems Science |year=2009 |publisher=Springer |isbn=978-0-387-30440-3 |pages=4960–4971 |chapter-url=https://link.springer.com/referenceworkentry/10.1007/978-0-387-30440-3_295 |language=en |chapter=Inverse Scattering Transform and the Theory of Solitons|doi=10.1007/978-0-387-30440-3_295 }}

Since <math>v\not=0</math>, this implies that <math>\frac{d\lambda}{dt}=0</math> [[iff|if and only if]]

* {{cite book |last1=Drazin |first1=P. G. |last2=Johnson |first2=R. S. |title=Solitons: An Introduction |year=1989 |publisher=Cambridge University Press |isbn=978-0-521-33655-0 |url=https://books.google.com/books?id=HPmbIDk2u-gC |language=en}}

:<math>\frac{dL}{dt} + LM - ML = 0. \, </math>

*{{cite journal |last1=Gardner |first1=Clifford S. |last2=Greene |first2=John M. |last3=Kruskal |first3=Martin D. |last4=Miura |first4=Robert M. |title=Method for Solving the Korteweg-deVries Equation |journal=Physical Review Letters |year=1967 |volume=19 |issue=19 |pages=1095–1097 |doi=10.1103/PhysRevLett.19.1095 |bibcode=1967PhRvL..19.1095G |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.19.1095|url-access=subscription }}

This is [[Lax's equation]]. One important thing to note about Lax's equation is that <math>\frac{dL}{dt}</math> is the time derivative of <math>L</math> precisely where it explicitly depends on <math>t</math>. The reason for defining the differentiation this way is motivated by the simplest instance of <math>L</math>, which is the Schrödinger operator (see [[Schrödinger equation]]):

*{{cite book |last1=Konopelchenko |first1=B.G. |last2=Dubrowsky |first2=V.G. |editor1-last=Sattinger |editor1-first=David H. |editor2-last=Tracy |editor2-first=C.A. |editor3-last=Venakides |editor3-first=Stephanos |title=Inverse Scattering and Applications |year=1991 |publisher=American Mathematical Soc. |isbn=978-0-8218-5129-6 |pages=77–90 |url=https://books.google.com/books?id=yTYcCAAAQBAJ |language=en |chapter=Localized solitons for the Ishimori equation}}

:<math>L=\frac{d^{2}}{dx^{2}}+u,</math>

*{{cite book |last1=Oono |first1=H. |editor1-last=Alfinito |editor1-first=E. |editor2-last=Boiti |editor2-first=M. |editor3-last=Martina |editor3-first=L. |title=Nonlinear Physics: Theory and Experiment |year=1996 |publisher=World Scientific Publishing Company Pte Limited |isbn=978-981-02-2559-9 |pages=241–248 |url=https://books.google.com/books?id=35EfzQEACAAJ |language=en |chapter=N-Soliton solution of Harry Dym equation by inverse scattering method.}}

where u is the "potential". Comparing the expression <math>\frac{dL}{dt}v+L\frac{dv}{dt}</math> with <math>\frac{\partial}{\partial t}\left(\frac{d^{2}v}{dx^{2}}+uv\right)</math> shows us that <math>\frac{\partial L}{\partial t}=\frac{du}{dt},</math> thus ignoring the first term.

*{{cite journal |last1=Osborne |first1=A. R. |title=Soliton physics and the periodic inverse scattering transform |journal=Physica D: Nonlinear Phenomena |year=1995 |volume=86 |issue=1 |pages=81–89 |doi=10.1016/0167-2789(95)00089-M |url=https://www.sciencedirect.com/science/article/abs/pii/016727899500089M |issn=0167-2789|url-access=subscription }}

After concocting the appropriate Lax pair it should be the case that Lax's equation recovers the original nonlinear PDE.

== Further reading ==

'''Step 3.''' Determine the time evolution of the eigenfunctions associated to each eigenvalue <math>\lambda</math>, 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.

*{{cite book |last1=Ablowitz |first1=Mark J. |last2=Clarkson |first2=P. A. |title=Solitons, Nonlinear Evolution Equations and Inverse Scattering |date=12 December 1991 |publisher=Cambridge University Press |isbn=978-0-521-38730-9 |url=https://books.google.com/books?id=RH44_EsqjGkC |language=en}}

*{{cite book |last1=Bullough |first1=R. K. |last2=Caudrey |first2=P. J. |title=Solitons |date=11 November 2013 |publisher=Springer Science & Business Media |isbn=978-3-642-81448-8 |url=https://books.google.com/books?id=kFT-CAAAQBAJ |language=en}}

'''Step 4.''' Perform the ''inverse scattering'' procedure by solving the [[Gelfand–Levitan–Marchenko integral equation]] ([[Israel Moiseevich Gelfand]] and [[Boris Moiseevich Levitan]];<ref>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.</ref> [[Vladimir Aleksandrovich Marchenko]]<ref>V. A. Marchenko, "Sturm-Liouville Operators and Applications", Birkhäuser, Basel, 1986.</ref>), 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 <math>L</math> 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.)

==List of integrable equations==

* [[Korteweg–de Vries equation]]

* [[nonlinear Schrödinger equation]]

* [[Camassa-Holm equation]]

* [[Sine-Gordon equation]]

* [[Toda lattice]]

* [[Ishimori equation]]

* [[Dym equation]] and so on.

==References==

{{reflist}}

*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.

*{{citation|mr=0336122|last1=Gardner|first1=Clifford S.|last2= Greene|first2= John M.|last3= Kruskal|first3= Martin D.|last4= Miura|first4= Robert M.|title=Korteweg-deVries equation and generalization. VI. Methods for exact solution.

|journal=Comm. Pure Appl. Math.|volume= 27 |year=1974|pages= 97–133|doi=10.1002/cpa.3160270108}}

*V. A. Marchenko, "Sturm-Liouville Operators and Applications", Birkhäuser, Basel, 1986.

*{{cite book |last1=Gelʹfand |first1=Izrailʹ Moiseevich |title=On the Determination of a Differential Equation from Its Spectral Function |date=1955 |publisher=American Mathematical Society |page=253-304|url=https://books.google.com/books?id=tSusYgEACAAJ |language=en}}

*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.

*{{cite book |last1=Marchenko |first1=Vladimir A. |title=Sturm-Liouville Operators and Applications |series=Operator Theory: Advances and Applications |date=1986 |volume=22 |___location=Basel|publisher=Birkhäuser|doi=10.1007/978-3-0348-5485-6 |isbn=978-3-0348-5486-3 |url=https://link.springer.com/book/10.1007/978-3-0348-5485-6 |language=en}}

*{{cite book |last1=Shaw |first1=J. K. |title=Mathematical Principles of Optical Fiber Communication |date=1 May 2004 |publisher=SIAM |isbn=978-0-89871-556-9 |url=https://books.google.com/books?id=GxpTjLWWAfcC |language=en}}

==External links==

* {{PDFlinkcite web|[url= http://www.math.uwaterloo.ca/~karigiannis/papers/ist.pdf |title=Introductory mathematical paper on IST] }}&nbsp;{{small|(300&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 307827 bytes -->)}}

* [httphttps://arxiv.org/abs/0905.4746 Inverse Scattering Transform and the Theory of Solitons]

{{Integrable systems}}

[[Category:Scattering theory]]

[[Category:Exactly solvable models]]

[[Category:Partial differential equations]]

[[Category:Transforms]]

[[Category:Integrable systems]]