20,067 research outputs found

    Diffractive Nonlinear Geometrical Optics for Variational Wave Equations and the Einstein Equations

    Full text link
    We derive an asymptotic solution of the vacuum Einstein equations that describes the propagation and diffraction of a localized, large-amplitude, rapidly-varying gravitational wave. We compare and contrast the resulting theory of strongly nonlinear geometrical optics for the Einstein equations with nonlinear geometrical optics theories for variational wave equations

    Paraxial light in a Cole-Cole nonlocal medium: integrable regimes and singularities

    Full text link
    Nonlocal nonlinear Schroedinger-type equation is derived as a model to describe paraxial light propagation in nonlinear media with different `degrees' of nonlocality. High frequency limit of this equation is studied under specific assumptions of Cole-Cole dispersion law and a slow dependence along propagating direction. Phase equations are integrable and they correspond to dispersionless limit of Veselov-Novikov hierarchy. Analysis of compatibility among intensity law (dependence of intensity on the refractive index) and high frequency limit of Poynting vector conservation law reveals the existence of singular wavefronts. It is shown that beams features depend critically on the orientation properties of quasiconformal mappings of the plane. Another class of wavefronts, whatever is intensity law, is provided by harmonic minimal surfaces. Illustrative example is given by helicoid surface. Compatibility with first and third degree nonlocal perturbations and explicit solutions are also discussed.Comment: 12 pages, 2 figures; eq. (36) corrected, minor change

    Semiclassical Asymptotics for the Maxwell - Dirac System

    Full text link
    We study the coupled system of Maxwell and Dirac equations from a semiclassical point of view. A rigorous nonlinear WKB-analysis, locally in time, for solutions of (critical) order O(ϵ)O(\sqrt{\epsilon}) is performed, where the small semiclassical parameter ϵ\epsilon denotes the microscopic/macroscopic scale ratio

    Norm-inflation for periodic NLS equations in negative Sobolev spaces

    Get PDF
    In this paper we consider Schr{\"o}dinger equations with nonlinearities of odd order 2σ\sigma + 1 on T^d. We prove that for σ\sigmad\ge2, they are strongly illposed in the Sobolev space H^s for any s \textless{} 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schr{\"o}dinger equation and its renormalized version we prove such a result for H^s with s \textless{} --2/3.Comment: 18 page
    corecore