Lie Brackets, Coupling and Riemann
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.
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
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
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.