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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 : with u(x,t) real.

So the solution depend on the 3 parameters et and the length of the interval . 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 , and which verify : The resolution of this system give the no-dimensionnal number : . 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 belonging to , 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 : This result is very useful, the rest fo the study will be realized for , and . We only change the value of . That simplify the study and the writing of the code to simulate the KSE.    Next: Stability of the solution Up: Theoretical study of the Previous: Theoretical study of the   Contents
Julien Delbove
2000-11-23