Reduction and reconstruction for self-similar dynamical systems

C. W. Rowley, I. G. Kevrekidis, J. E. Marsden, and K. Lust

Nonlinearity 16 : 12571275, August 2003.

Abstract:

We present a general method for analyzing 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 behavior 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.

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