Resonance in dispersive wave systems

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1999.Includes bibliographical references (p. 110-112).The focus of this thesis is to investigate the differences that arise in weakly nonlinear wave interactions under the assumption of a discrete or continuous spectrum. In particular the latter is investigated in detail for the case of three wave interactions. It is known that an extra condition on the group velocities is required for resonant growth. Such so called double resonances can be shown to occur in a variety of physical regimes. A direct multiple scale analysis of the spectral representation of a model equation containing arbitrary linear dispersion and weak quadratic nonlinearity was conducted. Consequently a system of "three wave" equations analogous to those for simple resonances was derived for the double resonance case. Key distinctions include an asymmetry between the temporal evolution of the modes and a longer time scale of ... for the case of a discrete simple triad resonance. A number of numerical simulations were then conducted for a variety of dispersions and nonlinearities in order to verify and extend the analytic results. Furthermore, a generalized version of the discrete three wave equations containing higher order dispersive terms was investigated with the intention of providing a link between the continuous and discrete three wave cases. Both analytic and numerical studies were conducted for a number of parameter regimes. In particular for the case analogous to the double resonance, energy propagation and transfer at the group velocity predicted by the continuous theory was seen. But differences also persisted in the time scales which reinforced the subtle, yet significant, distinction between the continuous and discrete points of view. Finally, a discussion of double resonances and their effect on statistical treatments of turbulent flows was given. The existence of double resonances appeared to effect the hierarchy of the perturbation expansions, and subsequent closures, in a significant fashion. A modified closure was proposed containing terms arising from both simple and double resonances.by David Embury Amundsen.Ph.D