We study the stability and instability of periodic traveling waves for Korteweg-de Vries type equations with fractional dispersion and related, nonlinear dispersive equations. We show that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations. We then discuss when the associated linearized equation admits solutions exponentiall...

We study the resolvent equation associated with a linear operator L arising from the linearized equation for perturbations of a steady Navier–Stokes flow U∗. We derive estimates which, together with a stability criterion from [33], show that the stability of U∗ (in the L–norm) depends only on the position of the eigenvalues of L, regardless the presence of the essential spectrum. Mathematics Su...

In this note we prove scattering for perturbations of solitons in the scaling space appropriate for the quartic nonlinearity, namely Ḣ− 1 6 . The article relies strongly on refined estimates for a KdV equation linearized at the soliton. In contrast to the work of Tao [19] we are able to work purely in the scaling space without additional regularity assumptions, allowing us to construct wave ope...

We construct a consistent model of gravity where the tensor graviton mode is massive, while linearized equations for scalar and vector metric perturbations are not modified. The Friedmann equation acquires an extra dark-energy component leading to accelerated expansion. The mass of the graviton can be as large as approximately (10(15) cm)(-1), being constrained by the pulsar timing measurements...

The quantum theory of linearized perturbations of the gravitational field of a Schwarzschild black hole is presented. The fundamental operators are seen to be the perturbed Weyl scalars Ψ̇0 and Ψ̇4 associated with the NewmanPenrose description of the classical theory. Formulae are obtained for the expectation values of the modulus squared of these operators in the Boulware, Unruh and Hartle-Hawki...

We consider the problem of asymptotic stability of a self-similar attractor for a simple semilinear radial wave equation which arises in the study of the Yang-Mills equations in 5+1 dimensions. Our analysis consists of two steps. In the first step we determine the spectrum of linearized perturbations about the attractor using a method of continued fractions. In the second step we demonstrate nu...

We establish pointwise bounds for the Green function and consequent linearized stability for multidimensional planar relaxation shocks of general relaxation systems whose equilibrium model is scalar, under the necessary assumption of spectral stability. Moreover, we obtain nonlinear L asymptotic behavior/sharp decay rate of perturbed weak shocks of general simultaneously symmetrizable relaxatio...

We calculate analytically low frequency quasi-normal modes of gravitational perturbations of AdS Schwarzschild black holes in d dimensions. We arrive at analytic expressions which are in agreement with their counterparts from linearized hydrodynamics in Sd−2×R, in accordance with the AdS/CFT correspondence. Our results are also in good agreement with results of numerical calculations. Research ...

The spiraling of adjacent trajectories in chaotic dynamical systems can be characterized by the distribution of local angular velocities of rotation of the displacement vector, which is governed by linearized equations of motion. This distribution, akin to that of local Lyapunov exponents, is studied for three examples of three-dimensional flows. Toy model shows that the rotation rate of adjace...

We show that the Gauss-Bonnet term is the only consistent curvature squared interaction in the Randall-Sundrum model and various static and inflationary solutions can be found. And from metric perturbations around the RS background with a single brane embedded, we also show that for a vanishing Gauss-Bonnet coefficient, the brane bending allows us to reproduce the 4D Einstein gravity at the lin...