On Landau damping

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of nonlinear echoes; sharp scattering estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the nonlinear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications.Comment: News: (1) the main result now covers Coulomb and Newton potentials, and (2) some classes of Gevrey data; (3) as a corollary this implies new results of stability of homogeneous nonmonotone equilibria for the gravitational Vlasov-Poisson equatio

On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture

On the well-posedness of the periodic KdV equation in high regularity classes

We prove well-posedness results for the initial value problem of the periodic KdV equation in classes of high regularity solutions. More precisely, we consider the problem in weighted Sobolev spaces, which comprise classical Sobolev spaces, Gevrey spaces, and analytic spaces. We show that the initial value problem is well posed in all spaces with subexponential growth of Fourier coefficients, and ‘almost well posed’ in spaces with exponential growth of Fourier coefficients

Appropriate Feedback in Asymmetric Interactions

Wrede B, Kopp S, Rohlfing K, Lohse M, Muhl C. Appropriate Feedback in Asymmetric Interactions. Journal of Pragmatics. 2010;42(9):2369-2384.Based on the observation that human-robot interaction is often laborious because the robot's interactional abilities fail to meet the user's expectations, we argue that feedback can play a central role in regulating expectations and mitigating unnecessary disruptions in the flow of conversation. For feedback to be appropriate in this sense, it needs to take situational information into account. This idea stems from interviews with persons with hearing and mental impairments who display perceptual limitations similar to a robot. The results of these interviews indicated that, depending on the goals of the situation, people with hearing impairments used either mediation (clarification) or concealment strategies to keep the interaction going. With this idea in mind, we analyzed human-robot interactions in two different situations - more task-oriented interactions versus more socially driven interactions - and we observed different feedback behaviors in users and their reactions to the robot's behavior. We use these results to derive a scaffold for modeling appropriate feedback in asymmetric interactions (i.e., in human-robot interactions) and briefly discuss some consequences for both the design of human-robot interaction and for theories of grounding. (c) 2010 Elsevier B.V. All rights reserved