3,266 research outputs found
Short-Time Existence for Scale-Invariant Hamiltonian Waves
We prove short-time existence of smooth solutions for a class of nonlinear,
and in general spatially nonlocal, Hamiltonian evolution equations that
describe the self-interaction of weakly nonlinear scale-invariant waves. These
equations include ones that describe weakly nonlinear hyperbolic surface waves,
such as nonlinear Rayleigh wave
Orientation Waves in a Director Field With Rotational Inertia
We study the propagation of orientation waves in a director field with
rotational inertia and potential energy given by the Oseen-Frank energy
functional from the continuum theory of nematic liquid crystals. There are two
types of waves, which we call splay and twist waves. Weakly nonlinear splay
waves are described by the quadratically nonlinear Hunter-Saxton equation.
Here, we show that weakly nonlinear twist waves are described by a new
cubically nonlinear, completely integrable asymptotic equation. This equation
provides a surprising representation of the Hunter-Saxton equation as an
advection equation. There is an analogous representation of the Camassa-Holm
equation. We use the asymptotic equation to analyze a one-dimensional initial
value problem for the director-field equations with twist-wave initial data
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
On a nonlocal analog of the Kuramoto-Sivashinsky equation
We study a nonlocal equation, analogous to the Kuramoto-Sivashinsky equation,
in which short waves are stabilized by a possibly fractional diffusion of order
less than or equal to two, and long waves are destabilized by a backward
fractional diffusion of lower order. We prove the global existence, uniqueness,
and analyticity of solutions of the nonlocal equation and the existence of a
compact attractor. Numerical results show that the equation has chaotic
solutions whose spatial structure consists of interacting traveling waves
resembling viscous shock profiles
Nonlinear surface plasmons
We derive an asymptotic equation for quasi-static, nonlinear surface plasmons
propagating on a planar interface between isotropic media. The plasmons are
nondispersive with a constant linearized frequency that is independent of their
wavenumber. The spatial profile of a weakly nonlinear plasmon satisfies a
nonlocal, cubically nonlinear evolution equation that couples its left-moving
and right-moving Fourier components. We prove short-time existence of smooth
solutions of the asymptotic equation and describe its Hamiltonian structure. We
also prove global existence of weak solutions of a unidirectional reduction of
the asymptotic equation. Numerical solutions show that nonlinear effects can
lead to the strong spatial focusing of plasmons. Solutions of the
unidirectional equation appear to remain smooth when they focus, but it is
unclear whether or not focusing can lead to singularity formation in solutions
of the bidirectional equation
Long time Solutions for a Burgers-Hilbert Equation via a Modified Energy Method
We consider an initial value problem for a quadratically nonlinear inviscid
Burgers-Hilbert equation that models the motion of vorticity discontinuities.
We use a modified energy method to prove the existence of small, smooth
solutions over cubically nonlinear time-scales
- …