We present a general method for analysing and numerically solving partial
differential equations with self-similar solutions. The method employs
ideas from symmetry reduction in geometric mechanics, and involves
separating the dynamics on the
shape space (which determines the overall
shape of the solution) from those on the
group space (which determines the
size and scale of the solution). The method is computationally tractable
as well, allowing one to compute self-similar solutions by evolving a dynamical
system to a steady state, in a scaled reference frame where the self-similarity
has been factored out. More generally, bifurcation techniques can be used to
find self-similar solutions, and determine their behaviour as parameters in the
equations are varied.
The method is given for an arbitrary Lie group, providing
equations for the dynamics on the reduced space, for reconstructing the full
dynamics and for determining the resulting scaling laws for self-similar solutions.
We illustrate the technique with a numerical example, computing self-similar
solutions of the Burgers equation.
