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In [[mathematics]], '''Bäcklund transforms''' or '''Bäcklund transformations''' (called 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== | |||
{{see|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 ''u'' and ''v'' of a [[holomorphic function]]. This first order system of partial differential equations has the following properties. | |||
# If ''u'' and ''v'' are solutions of the Cauchy–Riemann equations, then ''u'' is a solution of the [[Laplace equation]] | |||
: <math>u_{xx} + u_{yy} = 0</math> | |||
(i.e., a [[harmonic function]]), and so is ''v''. This follows straightforwardly by differentiating the equations with respect to ''x'' and ''y'' and using the fact that | |||
#:<math>u_{xy}=u_{yx}, \quad v_{xy}=v_{yx},.\,</math> | |||
# Conversely if ''u'' is a solution of Laplace's equation, then there exist functions ''v'' which solve the Cauchy–Riemann equations together with ''u''. | |||
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 ''u'' and Laplace's equation for ''v'' 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 ''u'', and a Bäcklund transform from ''u'' to ''v'', we can deduce a partial differential equation satisfied by ''v''. | |||
This example is rather trivial, because all three equations (the equation for ''u'', the equation for ''v'' 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== | |||
{{see|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{u+v}{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== | |||
{{see|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. | |||
== References == | |||
{{refbegin}} | |||
* {{Cite book | |||
| publisher = Math Sci Press | |||
| isbn = 978-0-915692-16-3 | |||
| last = Hermann | |||
| first = Robert | |||
| 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 | |||
| postscript =, [http://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 --> | |||
[[Category:Differential geometry]] | |||
[[Category:Solitons]] | |||
[[Category:Exactly solvable models]] | |||
[[Category:Surfaces]] | |||
[[Category:Transforms]] | |||
{{DEFAULTSORT:Backlund Transform}} | |||
Revision as of 22:22, 29 October 2012
In mathematics, Bäcklund transforms or Bäcklund transformations (called after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions. They are an important tool in soliton theory and integrable systems. 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
Bäcklund transforms have their origins in differential geometry: the first nontrivial example is the transformation of pseudospherical surfaces introduced by L. Bianchi and 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
Template:See The prototypical example of a Bäcklund transform is the Cauchy–Riemann system
which relates the real and imaginary parts u and v of a holomorphic function. This first order system of partial differential equations has the following properties.
- If u and v are solutions of the Cauchy–Riemann equations, then u is a solution of the Laplace equation
(i.e., a harmonic function), and so is v. This follows straightforwardly by differentiating the equations with respect to x and y and using the fact that
- Conversely if u is a solution of Laplace's equation, then there exist functions v which solve the Cauchy–Riemann equations together with u.
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 u and Laplace's equation for v are the integrability conditions for solving the Cauchy–Riemann equations.
These are the characteristic features of a Bäcklund transform. If we have a partial differential equation in u, and a Bäcklund transform from u to v, we can deduce a partial differential equation satisfied by v.
This example is rather trivial, because all three equations (the equation for u, the equation for v 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
Template:See Suppose that u is a solution of the sine-Gordon equation
Then the system
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
Template:See 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
where a is an arbitrary parameter, and if u is a solution of the Liouville equation
then v is a solution of the much simpler equation, , and vice versa.
We can then solve the (non-linear) Liouville equation by working with a much simpler linear equation.
References
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, 2004.
External links
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