Pattern formation in Hamiltonian systems with continuous spectra; a normalform singlewave model
Abstract
Pattern formation in biological, chemical and physical problems has received considerable attention, with much attention paid to dissipative systems. For example, the GinzburgLandau 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 "singlewave model" arises in many physical contexts that share common pattern forming behavior. These systems have Hamiltonian structure, and the singlewave model is a kind of Hamiltonian meanfield theory describing the patterns that form in phase space. The singlewave 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 singlewave model also arises in fluid mechanics, specifically shearflow 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 singlewave model emerges from a large class of nonlinear advectiontransport 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 (waveparticle 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 normalform character of the singlewave model.
 Publication:

arXiv eprints
 Pub Date:
 February 2013
 arXiv:
 arXiv:1303.0065
 Bibcode:
 2013arXiv1303.0065B
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Mathematical Physics;
 Nonlinear Sciences  Pattern Formation and Solitons