DispersiveWiki:Sandbox

From DispersiveWiki
Jump to navigationJump to search

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.



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 FraA2006.

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

The non trivial set of equations is so obtained

where . Indeed, this is a gradient expansion.

An interesting problem that applies to a given PDE is

where is a driving term. When a small parameter series has to be computed we obtain that at the leading order one has generally to solve

that is very easy to solve by the Green function method

where

and

being $\delta^D$ a Dirac distribution of the given dimensionality .

A similar result holds also for the large parameter series FraA2007,FraB2007. We note that the leading order of the gradient expansion is now

A small time series holds that has the form

being

and the coefficients are computed by deriving the equation we started from and with the initial conditions and will generally depend on the values of the source and its derivatives at the intial time. The success of the method relies on the ability to obtain analitically the Green function.

An interesting case is given by FraB2006. In this case one has

I

where is the step function, is a Jacobi elliptic function and is a scale parameter being the theory scale invariant. This gives immediately the mass spectrum of the quantum theory as FraB2006

where is an integer and .