Spatially localized states in the Zhang-Vinals equation

空间局域态通常由空间中有良好边界的区域中的图案组成，并被均匀的背景包围。这些现象在静止状态下是很好理解的，但对它们随时间变化的对应物却知之甚少。这里我们对振子感兴趣，这是一种空间局域状态，图案在其中振荡。在Lioubashevski等人的Phys中发现了一个明显的例子。莱特牧师。83,3190--3193(1999)。在本文中，复杂的流体被垂直振动，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，可以揭示这些结构，但幸运的是，金兹堡-朗道方程，已知的模型接近开始非常准确的动力学，正在为即将到来的工作提供宝贵的指导(参见A.S.Alnahdi，J.Niesen和A.M.Rucklidge)。Siam J.应用动力系统13(2014)1311-1327)。这个项目的想法是考虑一种称为张-维纳斯模型的振动流体层模型，并在上面关于金兹堡-朗道方程的工作的指导下，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(图案的形成)，以及一些渐近性(Ginzburg-Landau方程的推导)、分析方法(Zhang-Vinals模型的推导)和一些数值(积分Zhang-Vinals模型)。

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).