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Week 4

7/26 Week 7:

Lie Brackets, Coupling and Riemann Invarients

Sometimes a complicated wave interaction can occur. Using Taylor's theorem, the extent of the interaction can be determined. The conceptual tool we use to study this is called the "Lie Bracket." The Lie Bracket of two vectors v1 and v2 is defined to be

LB[v1,v2] = D[v1].v2-D[v2].v1.

The Lie Bracket of two waves tells us if two waves interact, and if they do, to what extent. If the Lie Bracket is zero, there is no interaction.

We represent a set of waves by the eigenvectors associated to those waves. A set of eigenvectors is "in involution" if the lie brackets of all pairs of eigenvectors are linear combinations of the eigenvectors in the set.

We go back and calculate the lie brackets of each system and we express each lie bracket as a linear combination of the right eigenvectors. We write a program that integrates the eigenvectors.

A Riemann invariant of a wave is a function of the state variables that does not change across that wave. Consider the left eigenvector associated to a wave. If a multiple of that eigenvector is has curl zero then that wave has a Riemann invariant. The derivative of the Riemann invariant is the left eigenvector.

Produced and maintained by Robert Chase, the Department of Mathematics and Statistics