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Diminution of the number of parameter

The Karamoto-Sivashinsky equation (KSE) is a one-dimensional partial derivative equation. The most usual form of KSE is :

$\displaystyle \frac{ \partial u}{\partial t}+ u \frac{ \partial u}{\partial x}+...
... \partial^2 u}{{\partial x}^2}+ \lambda \frac{ \partial^4 u}{{\partial x}^4}=0
$

with u(x,t) real.

So the solution depend on the 3 parameters $ \nu $ et $ \lambda $ and the length of the interval $ L $. The number of physical dimension is 2 ; there are a space dimension and a time one. As a conclusion of this remarks, the solution depends on a unique parameter.

The different dimensions are :

In order to determine the parameter on which the solution depend, we have to find a no-dimensionnal number. This is equivalent to find $ \alpha$, $ \beta$ and $ \gamma$ which verify :

$\displaystyle [L]^{\alpha} [\nu]^{\beta} [\lambda]^{\gamma}=[1]
$

The resolution of this system give the no-dimensionnal number : $ \frac{\lambda}{L^2 \nu}$. So the solution of the KSE depend of this number. This result simplify the study, in fact instead of simulate the KSE for each triplet $ (L,\nu,\lambda)$ belonging to $ R^3$, we just have to study its behaviour when the no-dimensionnal number change.

Another way to prouve that the solution of the KSE depends only on a no-dimensionnal parameter is to transform the dimensionnal equation to a no-dimensionnal one. We first have to dertermine the values of reference. There are different possibities, one of these is :

Then we consider that : We inject that in the KSE, and that lead to a new equation which depend on a unique parameter :

$\displaystyle \frac{ \partial u^*}{\partial t^*}+ u^* \frac{ \partial u^*}{\par...
...l x^*}^2}+ \frac{\lambda}{L^2 \nu} \frac{ \partial^4 u^*}
{{\partial x^*}^4}=0
$

This result is very useful, the rest fo the study will be realized for $ L=1$, and $ \lambda=1$. We only change the value of $ \nu $. That simplify the study and the writing of the code to simulate the KSE.


next up previous contents
Next: Stability of the solution Up: Theoretical study of the Previous: Theoretical study of the   Contents
Julien Delbove
2000-11-23