Spatially localized states in the Zhang-Vinals equation

空间定域状态状态通常由空间中边界良好的区域中的图案组成，并被均匀的背景包围。这些在静止状态下被很好地理解，但对它们的时间依赖性的对应物知之甚少。我们这里感兴趣的是振子，振子是空间定域状态，其中图形在振荡。在Lioubashevski等人的研究中发现了一个引人注目的例子。Rev. Lett. 83, 3190—3193（1999）。在本文中，复杂流体被垂直振动，并在非振荡背景中显示振荡峰。目前还没有关于流体方程的理论工作可以解释这些结构，但幸运的是，Ginzburg- Landau方程为即将到来的工作提供了宝贵的指导(参见A.S. Alnahdi， J. Niesen和A.M.Rucklidge。李建军，刘建军，李建军，等。这个项目的想法是考虑一个被称为Zhang- Vinals模型的振动流体层模型，并在上面的Ginzburg- Landau方程的工作指导下，找到并分析这些振荡的形成。这项工作将主要集中在构型的非线性方面（模式形成），包括一些渐近性（Ginzburg- Landau方程的推导）、解析方法（Zhang- Vinals模型的推导）和一些数值（对Zhang- Vinals模型进行积分）。

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).