We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any riemannian compact manifold. As a consequence we infer global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on three–manifolds in the case of quadratic nonlinearities. We also discuss the optimali...

We have synthesized a series of chalcogenide glasses from the As-S-Se system that is designed to have strong nonlinearities. Measurements reveal that many of these glasses offer optical Kerr nonlinearities greater than 400 times that of fused silica at 1.25 and 1.55mum and figures of merit for all-optical switching greater than 5 at 1.55mum .

Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to show that localized nonlinearities can support bound states with an arbitrary number solitons, and discuss other applications of interest to the field of nonlinear matter waves.

In this paper we study the quenching problem in nonlinear heat equations with power nonlinearities. For nonlinearities of power p < 0 and for an open set of slowly varying initial conditions we prove that the solutions will collapse in a finite time. We find the collapse profile and estimate the remainder.

This paper presents a systematic approach to the design of a robust dynamic state feedback controller using copies of the plant nonlinearities, which is based on the use of IQCs and minimax LQR control. The approach combines a linear state feedback guaranteed cost controller and copies of the plant nonlinearities to form a robust nonlinear controller.

The method of surrogates is one of the key concepts of nonlinear data analysis. Here, we demonstrate that commonly used algorithms for generating surrogates often fail to generate truly linear time series. Rather, they create surrogate realizations with Fourier phase correlations leading to nondetections of nonlinearities. We argue that reliable surrogates can only be generated, if one tests se...

We study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half-line with a nonlinear boundary condition,    ut = uxx − λ(u + 1) log(u + 1) (x, t) ∈ R+ × (0, T ), −ux(0, t) = (u + 1) log(u + 1)(0, t) t ∈ (0, T ), u(x, 0) = u0(x) x ∈ R+, with p, q, λ > 0. We describe in terms of p, q and λ when the solution is global in time and when it blows up in f...

It is demonstrated how to generate time series with tailored nonlinearities by inducing well-defined constraints on the Fourier phases. Correlations between the phase information of adjacent phases and (static and dynamic) measures of nonlinearities are established and their origin is explained. By applying a set of simple constraints on the phases of an originally linear and uncorrelated Gauss...