Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model

Pattern formation in biological, chemical and physical problems has received considerable attention, with much attention paid to dissipative systems. For example, the Ginzburg–Landau equation is a normal form that describes pattern formation due to the appearance of a single mode of instability in a wide variety of dissipative problems. In a similar vein, a certain “single-wave model” arises in many physical contexts that share common pattern forming behavior. These systems have Hamiltonian structure, and the single-wave model is a kind of Hamiltonian mean-field theory describing the patterns that form in phase space. The single-wave model was originally derived in the context of nonlinear plasma theory, where it describes the behavior near threshold and subsequent nonlinear evolution of unstable plasma waves. However, the single-wave model also arises in fluid mechanics, specifically shear-flow and vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter physics, and other Hamiltonian theories characterized by mean field interaction. We demonstrate, by a suitable asymptotic analysis, how the single-wave model emerges from a large class of nonlinear advection-transport theories. An essential ingredient for the reduction is that the Hamiltonian system has a continuous spectrum in the linear stability problem, arising not from an infinite spatial domain but from singular resonances along curves in phase space whereat wavespeeds match material speeds (wave-particle resonances in the plasma problem, or critical levels in fluid problems). The dynamics of the continuous spectrum is manifest as the phenomenon of Landau damping when the system is … Such dynamical phenomena have been rediscovered in different contexts, which is unsurprising in view of the normal-form character of the single-wave model.