Wave propagation phenomena from a spatiotemporal viewpoint: Resonances and bifurcations

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 eigenvalues 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.