On the Schr\"odinger equations with isotropic and anisotropic fourth-order dispersion

This paper deals with the Cauchy problem associated to the nonlinear fourth-order Schr\"odinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $i\partial _{t}u+\epsilon \Delta u+\delta A u+\lambda|u|^\alpha u=0,$ $x\in \mathbb{R}^{n},$ $t\in \mathbb{R},$ where $A$ represents either the operator $\Delta^2$ (isotropic dispersion) or $\sum_{i=1}^d\partial_{x_ix_ix_ix_i},\ 1\leq d<n$ (anisotropic dispersion), and $\alpha, \epsilon, \lambda$ are given real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, such as weak-$L^p$ spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation $(\epsilon=0)$ for which, the existence of self-similar solutions is obtained as consequence of his scaling invariance. In a second part, we investigate the vanishing second order dispersion limit in the framework of weak-$L^p$ spaces. We also analyze the convergence of the solutions for the nonlinear fourth-order Schr\"odinger equation $i\partial _{t}u+\epsilon \Delta u+\delta \Delta^2 u+\lambda|u|^\alpha u=0$, as $\epsilon$ goes to zero, in $H^2$-norm, to the solutions of the corresponding biharmonic equation $i\partial _{t}u+\delta \Delta^2 u+\lambda|u|^\alpha u=0$

An Integrated Neural Network-Event-Related Potentials Model of Temporal and Probability Context Effects on Event Categorization

We present a neural network that adapts and integrates several preexisting or new modules to categorize events in short term memory (STM), encode temporal order in working memory, evaluate timing and probability context in medium and long term memory. The model shows how processed contextual information modulates event recognition and categorization, focal attention and incentive motivation. The model is based on a compendium of Event Related Potentials (ERPs) and behavioral results either collected by the authors or compiled from the classical ERP literature. Its hallmark is, at the functional level, the interplay of memory registers endowed with widely different dynamical ranges, and at the structural level, the attempt to relate the different modules to known anatomical structures.INSERM; NATO; DGA/DRET (911470/A000/DRET/DS/DR

Learning Temporal Contexts and Priming-Preparation Modes for Pattern Recognition

The system presented here is based on neurophysiological and electrophysiological data. It computes three types of increasingly integrated temporal and probability contexts, in a bottom-up mode. To each of these contexts corresponds an increasingly specific top-down priming effect on lower processing stages, mostly pattern recognition and discrimination. Contextual learning of time intervals, events' temporal order or sequential dependencies and events' prior probability results from the delivery of large stimuli sequences. This learning gives rise to emergent properties which closely match the experimental data.Institut national de la santé et de la recherche médicale; Ministère de la Défense Nationale (DGA/DRET 911470/AOOO/DRET/DS/DR); Consejo Nacional de Ciencia y Tecnología (63462

The regularized Benjamin–Ono and BBM equations: Well-posedness and nonlinear stability

AbstractNonlinear stability of nonlinear periodic solutions of the regularized Benjamin–Ono equation and the Benjamin–Bona–Mahony equation with respect to perturbations of the same wavelength is analytically studied. These perturbations are shown to be stable. We also develop a global well-posedness theory for the regularized Benjamin–Ono equation in the periodic and in the line setting. In particular, we show that the Cauchy problem (in both periodic and nonperiodic case) cannot be solved by an iteration scheme based on the Duhamel formula for negative Sobolev indices