Editing Bäcklund transform

In [[mathematics]], '''Bäcklund transforms''' or '''Bäcklund transformations''' (named after the Swedish mathematician [[Albert Victor Bäcklund]]) relate [[partial differential equation]]s and their solutions. They are an important tool in [[soliton theory]] and [[integrable system]]s. A Bäcklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter. It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a Bäcklund transformation of the other.

A Bäcklund transform which relates solutions of the ''same'' equation is called an '''invariant Bäcklund transform''' or '''auto-Bäcklund transform'''.  If such a transform can be found, much can be deduced about the solutions of the equation especially if the Bäcklund transform contains a parameter. However, no systematic way of finding Bäcklund transforms is known.

== History ==
[[File:Pseudosphere.png|thumb|Bäcklund transforms originated as transformations of [[pseudosphere]]s in the 1880s.]]
Bäcklund transforms have their origins in [[differential geometry]]: the first nontrivial example is the transformation of [[pseudospherical surface]]s introduced by [[Luigi Bianchi|L. Bianchi]] and [[Albert Victor Bäcklund|A.V. Bäcklund]] in the 1880s. This is a geometrical construction of a new pseudospherical surface from an initial such surface using a solution of a [[linear differential equation]]. Pseudospherical surfaces can be described as solutions of the [[sine-Gordon equation]], and hence the Bäcklund transformation of surfaces can be viewed as a transformation of solutions of the sine-Gordon equation.

==The Cauchy–Riemann equations==
{{Further|Cauchy–Riemann equations}}
The prototypical example of a Bäcklund transform is the [[Cauchy–Riemann equations|Cauchy–Riemann system]]

:<math>u_x=v_y, \quad u_y=-v_x,\,</math>

which relates the real and imaginary parts <math>u</math> and <math>v</math> of a [[holomorphic function]]. This first order system of partial differential equations has the following properties.

# If <math>u</math> and <math>v</math> are solutions of the Cauchy–Riemann equations, then <math>u</math> is a solution of the [[Laplace equation]] <br /><math>u_{xx} + u_{yy} = 0</math> <br />(i.e., a [[harmonic function]]), and so is <math>v</math>. This follows straightforwardly by differentiating the equations with respect to <math>x</math> and <math>y</math> and using the fact that <br /><math>u_{xy}=u_{yx}, \quad v_{xy}=v_{yx}.\,</math>
# Conversely if <math>u</math> is a solution of Laplace's equation, then there exist functions <math>v</math> which solve the Cauchy–Riemann equations together with <math>u</math>.
Thus, in this case, a Bäcklund transformation of a harmonic function is just a [[conjugate harmonic function]]. The above properties mean, more precisely, that Laplace's equation for <math>u</math> and Laplace's equation for <math>v</math> are the [[integrability condition]]s for solving the Cauchy–Riemann equations.

These are the characteristic features of a Bäcklund transform. If we have a partial differential equation in <math>u</math>, and a Bäcklund transform from <math>u</math> to <math>v</math>, we can deduce a partial differential equation satisfied by <math>v</math>.

This example is rather trivial, because all three equations (the equation for <math>u</math>, the equation for <math>v</math> and the Bäcklund transform relating them) are linear. Bäcklund transforms are most interesting when just one of the three equations is linear.

==The sine-Gordon equation==
{{Further|sine-Gordon equation}}
Suppose that ''u'' is a solution of the [[sine-Gordon equation]]
:<math> u_{xy} = \sin u.\,</math>

Then the system
:<math>\begin{align}
v_x & = u_x + 2a \sin \Bigl( \frac{v+u}{2} \Bigr) \\
v_y & = -u_y + \frac{2}{a} \sin \Bigl( \frac{v-u}{2} \Bigr)
\end{align} \,\!</math>
where ''a'' is an arbitrary parameter, is solvable for a function ''v'' which will also satisfy the sine-Gordon equation. This is an example of an auto-Bäcklund transform.

By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation.

==The Liouville equation==
{{Further|Liouville equation}}
A Bäcklund transform can turn a non-linear partial differential equation into a simpler, linear, partial differential equation.

For example, if ''u'' and ''v'' are related via the Bäcklund transform

:<math>\begin{align}
v_x & = u_x + 2a \exp \Bigl( \frac{u+v}{2} \Bigr) \\
v_y & = -u_y - \frac{1}{a} \exp \Bigl( \frac{u-v}{2} \Bigr)
\end{align} \,\!</math>
where ''a'' is an arbitrary parameter, and if ''u'' is a solution of the [[Liouville equation]]

<math>u_{xy}=\exp u \,\!</math>

then ''v'' is a solution of the much simpler equation, <math>v_{xy}=0</math>, and vice versa.

We can then solve the (non-linear) Liouville equation by working with a much simpler linear equation.

== See also ==
* [[Integrable system]]
* [[Korteweg–de Vries equation]]
* [[Darboux transformation]]

== References ==
{{Refbegin}}
* {{Cite book
| publisher = Math Sci Press
| isbn = 978-0-915692-16-3
| last = Hermann
| first = Robert
| author-link=Robert Hermann (mathematician)
| title = The geometry of non-linear differential equations, Bäcklund transformations, and solitons
| date = 1976
}}
* {{Citation
| edition = 1st
| publisher = Academic Press
| isbn = 0-12-592850-5
| last1 = Rogers
| first1 = C.
| first2 = W.F. | last2 = Shadwick
| title = Bäcklund transformations and their applications
| date = 1982-05-12
}}
* {{Citation
| publisher = [[Cambridge University Press]]
| isbn = 978-0-521-01288-1
| last1 = Rogers
| first1 = C.
| first2 = Wolfgang Karl | last2 = Schief
| title = Bäcklund and Darboux transformations
| date = 2002
}}, [https://www.loc.gov/catdir/samples/cam031/2001043453.pdf excerpt]
* A. D. Polyanin and V. F. Zaitsev, ''Handbook of Nonlinear Partial Differential Equations'', Chapman & Hall/CRC Press, 2004.
{{Refend}}

== External links ==
* {{springer|title=Bäcklund transformation|id=p/b120010}}
* {{MathWorld | title = Bäcklund Transformation | urlname = BaecklundTransformation }} <!-- has good list of references -->

{{Authority control}}

{{DEFAULTSORT:Backlund Transform}}
[[Category:Differential geometry]]
[[Category:Solitons]]
[[Category:Exactly solvable models]]
[[Category:Surfaces]]
[[Category:Transforms]]
[[Category:Integrable systems]]