Sine-Gordon Solitons

Author of Web page:
Guillermo Avendaño Franco

Léalo en Español

From field theory the wave equation for mesons is

$\begin{displaymath} \nabla^{2}u(x,t)-\frac{\partial^2 u(x,t)}{\partial t^2}=m^2 u(x,t)+g^2 u^3(x,t), \end{displaymath}$

(1)

the right hand side is the addition of the two first terms of the sin series

$\begin{displaymath} {\rm sen}(x)=\sum^{\infty}_{n=0}(-1)^n\frac{x^{2n+1}}{(2n+1)!} \end{displaymath}$

taking

and $g=\sqrt{-1/6}$ , then the bidimensional equation takes the form

$\begin{displaymath} \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}-\frac{\partial^2 u}{\partial t^2}={\rm sen}(u). \end{displaymath}$

(2)

The non-linearity of Sine-Gordon Eq. (1) makes difficult to obtain an analytic solution. To find a numeric solution lets start limiting the working space:

$\begin{displaymath} -x_0<x<x_0 \ \ -y_0<y<y_0 \ \ t\geq 0, \end{displaymath}$

(3)

with

, additionally the spatial derivate is taken zero on the borders.

$\begin{displaymath} \frac{\partial u}{\partial x}(-x_0,y ,t)=\frac{\partial u}{\partial x}(x_0,y ,t)=0 \end{displaymath}$

(4)

$\begin{displaymath} \frac{\partial u}{\partial y}(x,-y_0 ,t)=\frac{\partial u}{\partial y}(x,y_0,t)=0. \end{displaymath}$

(5)

As initial condition the function:

is selected, and its derivate at time

$\begin{displaymath} u(x,y,t=0)=4 \tan^{-1}(e^{3-\sqrt{x^{2}+y^{2}}}) \end{displaymath}$

(6)

$\begin{displaymath} \frac{\partial u}{\partial x}(x,y ,t=0)=0. \end{displaymath}$

(7)

Space and time are discretized as:

Continuous	$\Rightarrow$	Discret
	$\rightarrow$

$\begin{displaymath} x = m\, \Delta x,\ \ y = l\, \Delta y,\ \ t = n\, \Delta t, \end{displaymath}$

(8)

with $\Delta x=\Delta y$
Time is expresed with a superscript:

$\begin{displaymath} u^{n}_{m,l}\equiv u(m\ \Delta x,l\ \Delta x,n\ \Delta t,). \end{displaymath}$

(9)

the usual expansions for second derivatives :

$\displaystyle u_{m,l}^{n+1}\simeq$	$\textstyle -$	$\displaystyle u^{n-1}_{m,l}+2\left[1-2\left( \frac{\Delta t}{\Delta x}\right)^{2}\right]\ u^{n}_{m,l}$	(10)
	$\textstyle +$	$\displaystyle \left( \frac{\Delta t}{\Delta x}\right)^{2}(u^{n}_{m+1,l}+u^{n}_{m-1,l}+u^{n}_{m,l+1}+u^{n}_{m,l-1})$	(11)
	$\textstyle -$	$\displaystyle \Delta t^{2} {\rm sen}\left[\frac{1}{4}(u^{n}_{m+1,l}+u^{n}_{m-1,l}+u^{n}_{m,l+1}+u^{n}_{m,l-1})\right],$	(12)