20,067 research outputs found
Diffractive Nonlinear Geometrical Optics for Variational Wave Equations and the Einstein Equations
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
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
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 is performed,
where the small semiclassical parameter denotes the
microscopic/macroscopic scale ratio
Norm-inflation for periodic NLS equations in negative Sobolev spaces
In this paper we consider Schr{\"o}dinger equations with nonlinearities of
odd order 2 + 1 on T^d. We prove that for d2, 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
- …