N.J. Balmforth, Institute for Fusion Studies, University of Texas, Austin, TX 78712
G.R. Ierley, Scripps Institute of Oceanography, University of California, San Diego, CA 92093-0225
R. Worthing, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
ABSTRACT:A study is presented of a one-dimensional, nonlinear partial differential equation that describes evolution of dispersive, long-wave instability. The solutions, under certain specific conditions, take the form of trains of well-separated pulses. The dynamics of such patterns of pulses is investigated using singular perturbation theory and with numerical simulation. These tools permit the formulation of a theory of pulse interaction, and enable the mapping out of the range of behavior in parameter space. There are regimes in which steady trains form; such states can be studied with the asymptotic, pulse-interaction theory. In other regimes, pulse trains are unstable to global, wave-like modes or {\its radiation}. This can precipitate more violent phenomena involving pulse creation, or generate oscillating states which may follow Shil'nikov's route to temporal chaos. The asymptotic theory is generalized to take some account of radiative dynamics. In the limit of small dispersion, steady trains largely cease to exist; the system follows various pathways to temporal complexity and typical bifurcation sequences are sketched out. The investigation guides us to a critical appraisal of the asymptotic theory and uncovers the wealth of different types of behavior present in the system.