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Difference between revisions of "Bäcklund transformation"

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m (moved Backlund transformation to Bäcklund transformation over redirect: accented title)
m (AUTOMATIC EDIT (latexlist): Replaced 34 formulas out of 34 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
 
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It was A.V. Bäcklund who, in 1873, 1880, 1882, and 1883 introduced transformations between pairs of surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b1200101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b1200102.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b1200103.png" /> such that the surface element
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b1200104.png" /></td> </tr></table>
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of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b1200105.png" /> is connected to the surface element
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It was A.V. Bäcklund who, in 1873, 1880, 1882, and 1883 introduced transformations between pairs of surfaces $\Sigma$, $\Sigma ^ { \prime }$ in $\mathbf{R} ^ { 3 }$ such that the surface element
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b1200106.png" /></td> </tr></table>
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\begin{equation*} \left\{ x _ { 1 } , x _ { 2 } , u , \frac { \partial u } { \partial x _ { 1 } } , \frac { \partial u } { \partial x _ { 2 } } \right\} \end{equation*}
  
of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b1200107.png" /> by four relations of the type
+
of $\Sigma$ is connected to the surface element
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b1200108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation*} \left\{ x _ { 1 } ^ { \prime } , x _ { 2 } ^ { \prime } , u ^ { \prime } , \frac { \partial u ^ { \prime } } { \partial x _ { 1 } ^ { \prime } } , \frac { \partial u ^ { \prime } } { \partial x _ { 2 } ^ { \prime } } \right\} \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b1200109.png" /> [[#References|[a1]]], [[#References|[a2]]]. Bäcklund transformations may be considered as generalized Lie–Bianchi transformations (cf. also [[Bianchi transformation|Bianchi transformation]]). J. Clairin and E. Goursat extended Bäcklund's results [[#References|[a1]]], [[#References|[a2]]].
+
of $\Sigma ^ { \prime }$ by four relations of the type
 +
 
 +
\begin{equation} \tag{a1} B _ { i } \left( x _ { 1 } , x _ { 2 } , u , \frac { \partial u } { \partial x _ { 1 } } , \frac { \partial u } { \partial x _ { 2 } } : x _ { 1 } ^ { \prime } , x _ { 2 } ^ { \prime } , u ^ { \prime } , \frac { \partial u ^ { \prime } } { \partial x _ { 1 } ^ { \prime } } , \frac { \partial u ^ { \prime } } { \partial x _ { 2 } ^ { \prime } } \right) = 0, \end{equation}
 +
 
 +
where $i = 1 , \ldots , 4$ [[#References|[a1]]], [[#References|[a2]]]. Bäcklund transformations may be considered as generalized Lie–Bianchi transformations (cf. also [[Bianchi transformation|Bianchi transformation]]). J. Clairin and E. Goursat extended Bäcklund's results [[#References|[a1]]], [[#References|[a2]]].
  
 
Application of the integrability condition
 
Application of the integrability condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
\begin{equation} \tag{a2} \frac { \partial ^ { 2 } u ^ { \prime } } { \partial x _ { 1 } ^ { \prime } \partial x _ { 2 } ^ { \prime } } - \frac { \partial ^ { 2 } u ^ { \prime } } { \partial x _ { 2 } ^ { \prime } \partial x _ { 1 } ^ { \prime } } = 0 \end{equation}
  
leads, under certain circumstances, either to a pair of third-order equations or to a single second-order equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001011.png" />. The analogous integrability condition on the unprimed quantities leads, again under appropriate conditions, to a pair of third-order equations or to a single second-order equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001012.png" />. In this case, implicit in the Bäcklund relations, (a1) is a mapping between the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001014.png" /> of generally distinct systems of partial differential equations. This can be used as follows: If the solution of the transformed equation or pair of equations is known, then the Bäcklund relations (a1) may be used to generate the solution of the original equation or pair of equations. If the equations are invariant under the Bäcklund transformation (a1), then the latter may be used to construct an infinite sequence of new solutions from a known trivial solution. Both types of Bäcklund transformations have important applications. Bäcklund transformations may also be used to link certain non-linear equations to canonical forms whose properties are well known.
+
leads, under certain circumstances, either to a pair of third-order equations or to a single second-order equation for $u$. The analogous integrability condition on the unprimed quantities leads, again under appropriate conditions, to a pair of third-order equations or to a single second-order equation for $u ^ { \prime }$. In this case, implicit in the Bäcklund relations, (a1) is a mapping between the solutions $u ( x _ { 1 } , x _ { 2 } )$, $u ^ { \prime } ( x _ { 1 } ^ { \prime } , x _ { 2 } ^ { \prime } )$ of generally distinct systems of partial differential equations. This can be used as follows: If the solution of the transformed equation or pair of equations is known, then the Bäcklund relations (a1) may be used to generate the solution of the original equation or pair of equations. If the equations are invariant under the Bäcklund transformation (a1), then the latter may be used to construct an infinite sequence of new solutions from a known trivial solution. Both types of Bäcklund transformations have important applications. Bäcklund transformations may also be used to link certain non-linear equations to canonical forms whose properties are well known.
  
 
The classical Bäcklund transformation (a1) can be generalized to include second-order derivatives. Thus, transformations of the type
 
The classical Bäcklund transformation (a1) can be generalized to include second-order derivatives. Thus, transformations of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} B _ { i } ( x _ { m } , u , u _ { m } , u _ { m n } : x _ { m } ^ { \prime } , u ^ { \prime } , u _ { m } ^ { \prime } , u _ { m n } ^ { \prime } ) = 0, \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001016.png" /></td> </tr></table>
+
\begin{equation*} i = 1 , \dots , 4 ,\: m , n = 1,2, \end{equation*}
  
 
are introduced. These extensions of the Bäcklund transformation can be used to obtain Bäcklund solutions for the [[Korteweg–de Vries equation|Korteweg–de Vries equation]].
 
are introduced. These extensions of the Bäcklund transformation can be used to obtain Bäcklund solutions for the [[Korteweg–de Vries equation|Korteweg–de Vries equation]].
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1) Consider the two non-linear ordinary differential equations
 
1) Consider the two non-linear ordinary differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001017.png" /></td> </tr></table>
+
\begin{equation*} \frac { d ^ { 2 } u } { d t ^ { 2 } } = \operatorname { sin } ( u ) , \quad \frac { d ^ { 2 } v } { d t ^ { 2 } } = \operatorname { sinh } ( v ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001019.png" /> are real-valued functions. Then
+
where $u$ and $v$ are real-valued functions. Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001020.png" /></td> </tr></table>
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\begin{equation*} \frac { d u } { d t } - i \frac { d v } { d t } = 2 e ^ { i \lambda } \operatorname { sin } \left( \frac { 1 } { 2 } ( u + i v ) \right) \end{equation*}
  
 
defines a Bäcklund transformation.
 
defines a Bäcklund transformation.
  
2) Consider the one-dimensional [[Sine-Gordon equation|sine-Gordon equation]] in light-cone coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001022.png" />:
+
2) Consider the one-dimensional [[Sine-Gordon equation|sine-Gordon equation]] in light-cone coordinates $\xi $ and $ \eta $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001023.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial ^ { 2 } u } { \partial \xi \partial \eta } = \operatorname { sin } ( u ). \end{equation*}
  
 
Then
 
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001024.png" /></td> </tr></table>
+
\begin{equation*} \xi ^ { \prime } ( \xi , \eta ) = \xi , \quad \eta ^ { \prime } ( \xi , \eta ) = \eta, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001025.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial u ^ { \prime } ( \xi ^ { \prime } ( \xi , \eta ) , \eta ^ { \prime } ( \xi , \eta ) ) } { \partial \xi ^ { \prime } } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001026.png" /></td> </tr></table>
+
\begin{equation*} = \frac { \partial u } { \partial \xi } - 2 \lambda \operatorname { sin } ( \frac { u ( \xi , \eta ) + u ^ { \prime } ( \xi ^ { \prime } ( \xi , \eta ) , \eta ^ { \prime } ( \xi , \eta ) ) } { 2 } ), \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001027.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial u ^ { \prime } ( \xi ^ { \prime } ( \xi , \eta ) , \eta ^ { \prime } ( \xi , \eta ) ) } { \partial \eta ^ { \prime } } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001028.png" /></td> </tr></table>
+
\begin{equation*} = - \frac { \partial u } { \partial \eta } + \frac { 2 } { \lambda } \operatorname { sin } \left( \frac { u ( \xi , \eta ) - u ^ { \prime } ( \xi ^ { \prime } ( \xi , \eta ) , \eta ^ { \prime } ( \xi , \eta ) ) } { 2 } \right), \end{equation*}
  
defines an auto-Bäcklund transformation, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001029.png" /> is a non-zero real parameter. A non-linear superposability principle can be given, whereby an infinite sequence of solutions may be constructed by purely algebraic means. A hierarchy of solutions can be found starting from the trivial solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001030.png" />.
+
defines an auto-Bäcklund transformation, where $\lambda$ is a non-zero real parameter. A non-linear superposability principle can be given, whereby an infinite sequence of solutions may be constructed by purely algebraic means. A hierarchy of solutions can be found starting from the trivial solution $u ( \xi , \eta ) = 0$.
  
 
3) The [[Korteweg–de Vries equation|Korteweg–de Vries equation]]
 
3) The [[Korteweg–de Vries equation|Korteweg–de Vries equation]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001031.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial u } { \partial t } + 6 u \frac { \partial u } { \partial x } + \frac { \partial ^ { 3 } u } { \partial x ^ { 3 } } = 0 \end{equation*}
  
 
and the modified Korteweg–de Vries equation
 
and the modified Korteweg–de Vries equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001032.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial v } { \partial t } - 6 v ^ { 2 } \frac { \partial v } { \partial x } + \frac { \partial ^ { 3 } v } { \partial x ^ { 3 } } = 0 \end{equation*}
  
 
are related by the Miura transformation
 
are related by the Miura transformation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001033.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial v } { \partial x } = u + v ^ { 2 }, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120010/b12001034.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial v } { \partial t } = - \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - 2 \left( v \frac { \partial u } { \partial x } + u \frac { \partial v } { \partial x } \right). \end{equation*}
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Rogers,  W.F. Shadwick,  "Bäcklund transformations and their applications" , Acad.  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.L. Anderson,  N.H. Ibragimov,  "Lie–Bäcklund transformations in applications" , SIAM (Soc. Industrial Applied Math.)  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.-H. Steeb,  "Continuous symmetries, Lie algebras, differential equations and computer algebra" , World Sci.  (1996)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.-H. Steeb,  "Problems and solutions in theoretical and mathematical physics: Advanced problems" , '''II''' , World Sci.  (1996)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W.-H. Steeb,  N. Euler,  "Nonlinear evolution equations and Painlevé test" , World Sci.  (1988)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  C. Rogers,  W.F. Shadwick,  "Bäcklund transformations and their applications" , Acad.  (1982)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R.L. Anderson,  N.H. Ibragimov,  "Lie–Bäcklund transformations in applications" , SIAM (Soc. Industrial Applied Math.)  (1979)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  W.-H. Steeb,  "Continuous symmetries, Lie algebras, differential equations and computer algebra" , World Sci.  (1996)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  W.-H. Steeb,  "Problems and solutions in theoretical and mathematical physics: Advanced problems" , '''II''' , World Sci.  (1996)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  W.-H. Steeb,  N. Euler,  "Nonlinear evolution equations and Painlevé test" , World Sci.  (1988)</td></tr></table>

Latest revision as of 17:02, 1 July 2020

It was A.V. Bäcklund who, in 1873, 1880, 1882, and 1883 introduced transformations between pairs of surfaces $\Sigma$, $\Sigma ^ { \prime }$ in $\mathbf{R} ^ { 3 }$ such that the surface element

\begin{equation*} \left\{ x _ { 1 } , x _ { 2 } , u , \frac { \partial u } { \partial x _ { 1 } } , \frac { \partial u } { \partial x _ { 2 } } \right\} \end{equation*}

of $\Sigma$ is connected to the surface element

\begin{equation*} \left\{ x _ { 1 } ^ { \prime } , x _ { 2 } ^ { \prime } , u ^ { \prime } , \frac { \partial u ^ { \prime } } { \partial x _ { 1 } ^ { \prime } } , \frac { \partial u ^ { \prime } } { \partial x _ { 2 } ^ { \prime } } \right\} \end{equation*}

of $\Sigma ^ { \prime }$ by four relations of the type

\begin{equation} \tag{a1} B _ { i } \left( x _ { 1 } , x _ { 2 } , u , \frac { \partial u } { \partial x _ { 1 } } , \frac { \partial u } { \partial x _ { 2 } } : x _ { 1 } ^ { \prime } , x _ { 2 } ^ { \prime } , u ^ { \prime } , \frac { \partial u ^ { \prime } } { \partial x _ { 1 } ^ { \prime } } , \frac { \partial u ^ { \prime } } { \partial x _ { 2 } ^ { \prime } } \right) = 0, \end{equation}

where $i = 1 , \ldots , 4$ [a1], [a2]. Bäcklund transformations may be considered as generalized Lie–Bianchi transformations (cf. also Bianchi transformation). J. Clairin and E. Goursat extended Bäcklund's results [a1], [a2].

Application of the integrability condition

\begin{equation} \tag{a2} \frac { \partial ^ { 2 } u ^ { \prime } } { \partial x _ { 1 } ^ { \prime } \partial x _ { 2 } ^ { \prime } } - \frac { \partial ^ { 2 } u ^ { \prime } } { \partial x _ { 2 } ^ { \prime } \partial x _ { 1 } ^ { \prime } } = 0 \end{equation}

leads, under certain circumstances, either to a pair of third-order equations or to a single second-order equation for $u$. The analogous integrability condition on the unprimed quantities leads, again under appropriate conditions, to a pair of third-order equations or to a single second-order equation for $u ^ { \prime }$. In this case, implicit in the Bäcklund relations, (a1) is a mapping between the solutions $u ( x _ { 1 } , x _ { 2 } )$, $u ^ { \prime } ( x _ { 1 } ^ { \prime } , x _ { 2 } ^ { \prime } )$ of generally distinct systems of partial differential equations. This can be used as follows: If the solution of the transformed equation or pair of equations is known, then the Bäcklund relations (a1) may be used to generate the solution of the original equation or pair of equations. If the equations are invariant under the Bäcklund transformation (a1), then the latter may be used to construct an infinite sequence of new solutions from a known trivial solution. Both types of Bäcklund transformations have important applications. Bäcklund transformations may also be used to link certain non-linear equations to canonical forms whose properties are well known.

The classical Bäcklund transformation (a1) can be generalized to include second-order derivatives. Thus, transformations of the type

\begin{equation} \tag{a3} B _ { i } ( x _ { m } , u , u _ { m } , u _ { m n } : x _ { m } ^ { \prime } , u ^ { \prime } , u _ { m } ^ { \prime } , u _ { m n } ^ { \prime } ) = 0, \end{equation}

\begin{equation*} i = 1 , \dots , 4 ,\: m , n = 1,2, \end{equation*}

are introduced. These extensions of the Bäcklund transformation can be used to obtain Bäcklund solutions for the Korteweg–de Vries equation.

The jet-bundle formulation of Bäcklund transformations provides the best framework for a unified treatment of the subject [a1], [a3]. A number of interesting ordinary and partial differential equations admit Bäcklund and auto-Bäcklund transformations. They have been found to have applications in mathematical physics.

Examples of Bäcklund transformations.

See also [a1], [a3], [a4], [a5].

1) Consider the two non-linear ordinary differential equations

\begin{equation*} \frac { d ^ { 2 } u } { d t ^ { 2 } } = \operatorname { sin } ( u ) , \quad \frac { d ^ { 2 } v } { d t ^ { 2 } } = \operatorname { sinh } ( v ), \end{equation*}

where $u$ and $v$ are real-valued functions. Then

\begin{equation*} \frac { d u } { d t } - i \frac { d v } { d t } = 2 e ^ { i \lambda } \operatorname { sin } \left( \frac { 1 } { 2 } ( u + i v ) \right) \end{equation*}

defines a Bäcklund transformation.

2) Consider the one-dimensional sine-Gordon equation in light-cone coordinates $\xi $ and $ \eta $:

\begin{equation*} \frac { \partial ^ { 2 } u } { \partial \xi \partial \eta } = \operatorname { sin } ( u ). \end{equation*}

Then

\begin{equation*} \xi ^ { \prime } ( \xi , \eta ) = \xi , \quad \eta ^ { \prime } ( \xi , \eta ) = \eta, \end{equation*}

\begin{equation*} \frac { \partial u ^ { \prime } ( \xi ^ { \prime } ( \xi , \eta ) , \eta ^ { \prime } ( \xi , \eta ) ) } { \partial \xi ^ { \prime } } = \end{equation*}

\begin{equation*} = \frac { \partial u } { \partial \xi } - 2 \lambda \operatorname { sin } ( \frac { u ( \xi , \eta ) + u ^ { \prime } ( \xi ^ { \prime } ( \xi , \eta ) , \eta ^ { \prime } ( \xi , \eta ) ) } { 2 } ), \end{equation*}

\begin{equation*} \frac { \partial u ^ { \prime } ( \xi ^ { \prime } ( \xi , \eta ) , \eta ^ { \prime } ( \xi , \eta ) ) } { \partial \eta ^ { \prime } } = \end{equation*}

\begin{equation*} = - \frac { \partial u } { \partial \eta } + \frac { 2 } { \lambda } \operatorname { sin } \left( \frac { u ( \xi , \eta ) - u ^ { \prime } ( \xi ^ { \prime } ( \xi , \eta ) , \eta ^ { \prime } ( \xi , \eta ) ) } { 2 } \right), \end{equation*}

defines an auto-Bäcklund transformation, where $\lambda$ is a non-zero real parameter. A non-linear superposability principle can be given, whereby an infinite sequence of solutions may be constructed by purely algebraic means. A hierarchy of solutions can be found starting from the trivial solution $u ( \xi , \eta ) = 0$.

3) The Korteweg–de Vries equation

\begin{equation*} \frac { \partial u } { \partial t } + 6 u \frac { \partial u } { \partial x } + \frac { \partial ^ { 3 } u } { \partial x ^ { 3 } } = 0 \end{equation*}

and the modified Korteweg–de Vries equation

\begin{equation*} \frac { \partial v } { \partial t } - 6 v ^ { 2 } \frac { \partial v } { \partial x } + \frac { \partial ^ { 3 } v } { \partial x ^ { 3 } } = 0 \end{equation*}

are related by the Miura transformation

\begin{equation*} \frac { \partial v } { \partial x } = u + v ^ { 2 }, \end{equation*}

\begin{equation*} \frac { \partial v } { \partial t } = - \frac { \partial ^ { 2 } u } { \partial x ^ { 2 } } - 2 \left( v \frac { \partial u } { \partial x } + u \frac { \partial v } { \partial x } \right). \end{equation*}

References

[a1] C. Rogers, W.F. Shadwick, "Bäcklund transformations and their applications" , Acad. (1982)
[a2] R.L. Anderson, N.H. Ibragimov, "Lie–Bäcklund transformations in applications" , SIAM (Soc. Industrial Applied Math.) (1979)
[a3] W.-H. Steeb, "Continuous symmetries, Lie algebras, differential equations and computer algebra" , World Sci. (1996)
[a4] W.-H. Steeb, "Problems and solutions in theoretical and mathematical physics: Advanced problems" , II , World Sci. (1996)
[a5] W.-H. Steeb, N. Euler, "Nonlinear evolution equations and Painlevé test" , World Sci. (1988)
How to Cite This Entry:
Bäcklund transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=B%C3%A4cklund_transformation&oldid=50437
This article was adapted from an original article by W.-H. Steeb (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article