DispersiveWiki:Sandbox: Difference between revisions

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<math>
<math>
     u = \sum_n \lambda^n u_n
     u = \sum_{n=0}^{\infty} \lambda^n u_n
</math>
</math>


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<math>
<math>
     v = \sum_n\frac{1}{\lambda}v_n
     v = \sum_{n=0}^{\infty}\frac{1}{\lambda}v_n
</math>
</math>


is obtained by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion holding in the limit <math>\lambda\rightarrow\infty</math> dual to the small perturbation theory <math>\lambda\rightarrow 0</math> we started with and having an adiabatic equation at the leading order.
is obtained by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion holding in the limit <math>\lambda\rightarrow\infty</math> dual to the small perturbation theory <math>\lambda\rightarrow 0</math> we started with and having an adiabatic equation at the leading order.

Revision as of 08:24, 14 June 2007

Welcome to the sandbox! Please feel free to edit this page as you please by clicking on the "edit" tab at the top of this page. Terry 14:58, 30 July 2006 (EDT)

Some basic editing examples

  • You can create a link by enclosing a word or phrase in double brackets. Example: [[well-posed]] => well-posed
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this is the sandbox.


Duality in perturbation theory

Duality in perturbation theory has been introduced in Fra1998. It can be formulated by saying that a solution series with a large parameter is dual to a solution series with a small parameter as it can be obtained by interchanging the choice of the perturbation term in the given equation.

A typical perturbation problem can be formulated with the equation

being an arbitrary ordering parameter. A solution series with a small parameter can be computed taking

giving the following equations to be solved

where a derivative with respect to the ordering parameter is indicated by a prime. The choice of the ordering parameter is just a conventional matter and one can choice to consider as a perturbation instead with respect to the same parameter. Indeed one formally could write the set of equations

where and are interchanged with the new solution . In order to undertsand the expansion parameter we rescale the time variable as into the equation to be solved obtaining

and we introduce the small parameter . One sees that applying again the small perturbation theory to the parameter we get the required set of equations but now the time is scaled as , that is, at the leading order the development parameter of the series will enter into the scale of the time evolution producing a proper slowing down ruled by the equation

that is an equation for adiabatic evolution that in the proper limit will give the static solution . So, the dual series

is obtained by simply interchanging the terms for doing perturbation theory. This is a strong coupling expansion holding in the limit dual to the small perturbation theory we started with and having an adiabatic equation at the leading order.