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 :

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

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 :

• $[u]=LT^{-1}$
• $[L]=L$
• $[\nu]=L^2T^{-1}$
• $[\lambda]=L^4T^{-1}$

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 :

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

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 :

• $u_{ref}=\frac{\nu}{L}$
• $L_{ref}=L$
• $T_{ref}=\frac{L^2}{\nu}$

Then we consider that :

• $u=u_{ref} u^*$
• $L=L_{ref} L^*$
• $T=T_{ref} T^*$

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

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

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.