The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions

Fundamental global similarity solutions of the standard form u_\g(x,t)=t^{-\a_\g} f_\g(y), with the rescaled variable y= x/{t^{\b_\g}}, \b_\g= \frac {1-n \a_\g}{10}, where \a_\g>0 are real nonlinear eigenvalues (\g is a multiindex in R^N) of the tenth-order thin film equation (TFE-10) u_{t} = \nabla \cdot(|u|^{n} \n \D^4 u) in R^N \times R_+, n>0, are studied. The present paper continues the study began by the authors in the previous paper P. Alvarez-Caudevilla, J.D.Evans, and V.A. Galaktionov, The Cauchy problem for a tenth-order thin film equation I. Bifurcation of self-similar oscillatory fundamental solutions, Mediterranean Journal of Mathematics, No. 4, Vol. 10 (2013), 1759-1790. Thus, the following questions are also under scrutiny: (I) Further study of the limit n \to 0, where the behaviour of finite interfaces and solutions as y \to infinity are described. In particular, for N=1, the interfaces are shown to diverge as follows: |x_0(t)| \sim 10 \left( \frac{1}{n}\sec\left( \frac{4\pi}{9} \right) \right)^{\frac 9{10}} t^{\frac 1{10}} \to \infty as n \to 0^+. (II) For a fixed n \in (0, \frac 98), oscillatory structures of solutions near interfaces. (III) Again, for a fixed n \in (0, \frac 98), global structures of some nonlinear eigenfunctions \{f_\g\}_{|\g| \ge 0} by a combination of numerical and analytical methods

Modeling Emmetropization in an Incessantly Moving Eye

Many questions remain unanswered regarding the specific cues and mechanisms for emmetropization, the process by which, during development, the eye adjusts itself so that distant objects are in focus. Research has so far primarily focused on the spatial cues present in the image on the retina, such as the degree of blur. However, eye movements incessantly transform a mostly static scene into temporal modulations, so that the input to the retina is not an image, but a spatiotemporal flow of luminance. Models of retinal input signals indicate that this space-time reformatting caused by eye movements yields additional cues to the ones traditionally considered for emmetropization. These cues are implicit in the temporal modulations impinging onto retinal receptors and depend on the optics and shape of the eye and the spatial statistics of the visual scene. Here we examine the characteristics of these signals in the presence of normal eye movements and model the possible consequences of abnormal oculomotor behavior in the development of myopia and hyperopia