DispersiveWiki:Sandbox: Difference between revisions

From DispersiveWiki
Jump to navigationJump to search
Line 96: Line 96:


The choice of the perturbation term to compute a small or a large parameter series depends also on the way the derivatives of the field are managed.
The choice of the perturbation term to compute a small or a large parameter series depends also on the way the derivatives of the field are managed.
In order to see this we apply the computation given in the previous section by rescaling time as <math>t\rightarrow \lambda t</math> and take a solution series with a large parameter
\phi = \sum_{n=0}^{\infty}\frac{1}{\lambda^n}\phi_n.
The non trivial set of equations is so obtained
<math>\partial_t^2\phi_0+\lambda V(\phi)=0</math>
<math>\partial_t^2\phi_1+V'(\phi_0)\phi=\Delta^2\phi_0</math>
<math>\vdots</math>

Revision as of 08:44, 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
  • You can italicize using double apostrophes, and boldface using triple apostrophes. Examples: ''ad hoc'' => ad hoc; '''Miura transform''' => Miura transform.
  • LaTeX-style equations can be created using the <math> and </math> tags. Example: <math>M(u(t)) = \int_{\R^d} |u(t,x)|^2\ dx</math> => .
  • Bulleted un-numbered lists (like this one) can be created by placing an asterisk * at the beginning of each item. Numbered lists are similar but use #. One can nest lists using ** and ##, etc.
  • Create new sections using two equality signs = on each side of the section name (edit this sandbox for some examples).
  • You can sign your name using three or four tildes: ~~~ or ~~~~.

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.

The main mathematical problem of this kind of methods is the existence of the solution series. For the most interesting cases this series are not converging and represent asymptotic approximations to the true solution.

Finally, the success of this method relies on the possibility to obtain a proper analytical solution to the leading order equation.

Nonlinear PDE and Perturbation Methods

The application of the perturbation methods described above to PDE gives an interesting result, i.e. the dual series to the small parameter solution series is a gradient expansion Fra2006.

This can be seen by considering a NLKG equation

The choice of the perturbation term to compute a small or a large parameter series depends also on the way the derivatives of the field are managed.

In order to see this we apply the computation given in the previous section by rescaling time as and take a solution series with a large parameter

\phi = \sum_{n=0}^{\infty}\frac{1}{\lambda^n}\phi_n.

The non trivial set of equations is so obtained