Journal of Statistical Physics, Vol. 76, Nos. 3/4, 1994, p. 1005-1043.
By using biorthogonal decompositions, we show how uniformly propagating
waves, togehter with their velocity, shape, and amplitude, can be extracted from
a spatiotemporal signal consisting of the superposition of various traveling
waves. The interaction between the different waves manifests itself in space-time
resonances in case of a discrete biorthogonal spectrum and in resonant
wavepackets in case of a continuous biorthogonal spectrum. Resonances appear
as invariant subspaces under the biorthogonal operator, which leads to closed
sets of algebraic equations. The analysis is then extended to superpositions of
dispersive waves for which the (Fourier) dispersion relation is no longer linear.
We then show how a space-time bifurcation, namely a qualitative change in the
spatiotemporal nature of the solution, occurs when the biorthogonal operator is
a nonholomorphic function of a parameter. This takes place when two eigen-
values are degenerate in the biorthogonal spectrum and when the spatial and
temporal eigenvectors rotate within each eigenspace. Such a scenario applied
to the superposition of traveling waves leads to the generation of additional
waves propagating at new velocities, which can be computed from the spatial
and temporal eigenmodes involved in the process (namely the shape of the
propagating waves slightly before the bifurcation). An eigenvalue degeneracy,
however, does not necessarily lead to a bifurcation, a situation we refer to as
being self-avoiding. We illustrate our theoretical predictions by giving examples
of bifurcating and self-avoiding events in propagating phenomena.