Spatially localized states in the Zhang-Vinals equation

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Spatially localized states states are typically comprised of a pattern in a well-bounded region in space and surrounded by a homogeneous background. These are well understood in the stationary regime but very little is known about their time-dependent counterparts. We are here interested in oscillons, which are spatially localized states in which the pattern is oscillating. A striking example of this is found in Lioubashevski et al., Phys. Rev. Lett. 83, 3190--3193 (1999). In this paper, a complex fluid is vibrated vertically and displays oscillating peaks in the midst of a non-oscillating background. There is no theoretical work available on the fluid equations that could shed light on these structures but fortunately, the Ginzburg--Landau equation, known to model the dynamics close to onset with great accuracy, is giving precious guidance for upcoming work (see A.S. Alnahdi, J. Niesen and A.M. Rucklidge. SIAM J. Applied Dynamical Systems 13 (2014) 1311-1327).The idea for this project is to consider a model of a vibrating fluid layer known as the Zhang--Vinals model and, guided by the work on the Ginzburg--Landau equation above, find and analyse the formation of these oscillons. The work would focus mainly on the nonlinear aspects of the configuration (pattern formation), with some asymptotics (derivation of the Ginzburg--Landau equation), analytical methods (derivation of the Zhang--Vinals model) and some numerics (integrating the Zhang--Vinals model).