Self similar solutions in shallow water equations

Abstract

Infinitesimal transformations: Consider a one-parameter Liegroup of transformations x = X(x; ) with identity = 0 and lawof composition . If we expand x = X(x; ) about = 0 we getx = x + ( @X@ )=0 + 22 ( @2X@2 )=0 + · · ·x = x + ( @X@ )=0 + O(2)= x + (x)where (x) = ( @X@ )=0. This is called infinitesimaltransformation of x = X(x; ) and the components of (x) arecalled infinitesimals of x = X(x; ).