4,899 research outputs found
Geodesic Flow on the Diffeomorphism Group of the circle
We show that certain right-invariant metrics endow the infinite-dimensional
Lie group of all smooth orientation-preserving diffeomorphisms of the circle
with a Riemannian structure. The study of the Riemannian exponential map allows
us to prove infinite-dimensional counterparts of results from classical
Riemannian geometry: the Riemannian exponential map is a smooth local
diffeomorphism and the length-minimizing property of the geodesics holds.Comment: 15 page
On the Cauchy problem for a nonlinearly dispersive wave equation
We establish the local well-posedness for a new nonlinearly dispersive wave
equation and we show that the equation has solutions that exist for indefinite
times as well as solutions which blowup in finite times. Furthermore, we derive
an explosion criterion for the equation and we give a sharp estimate from below
for the existence time of solutions with smooth initial data.Comment: arxiv version is already officia
Equations of the Camassa-Holm Hierarchy
The squared eigenfunctions of the spectral problem associated with the
Camassa-Holm (CH) equation represent a complete basis of functions, which helps
to describe the inverse scattering transform for the CH hierarchy as a
generalized Fourier transform (GFT). All the fundamental properties of the CH
equation, such as the integrals of motion, the description of the equations of
the whole hierarchy, and their Hamiltonian structures, can be naturally
expressed using the completeness relation and the recursion operator, whose
eigenfunctions are the squared solutions. Using the GFT, we explicitly describe
some members of the CH hierarchy, including integrable deformations for the CH
equation. We also show that solutions of some - dimensional members of
the CH hierarchy can be constructed using results for the inverse scattering
transform for the CH equation. We give an example of the peakon solution of one
such equation.Comment: 10 page
On periodic water waves with Coriolis effects and isobaric streamlines
In this paper we prove that solutions of the f-plane approximation for
equatorial geophysical deep water waves, which have the property that the
pressure is constant along the streamlines and do not possess stagnation
points,are Gerstner-type waves. Furthermore, for waves traveling over a flat
bed, we prove that there are only laminar flow solutions with these properties.Comment: To appear in Journal of Nonlinear Mathematical Physics; 15 page
Particle trajectories in linearized irrotational shallow water flows
We investigate the particle trajectories in an irrotational shallow water
flow over a flat bed as periodic waves propagate on the water's free surface.
Within the linear water wave theory, we show that there are no closed orbits
for the water particles beneath the irrotational shallow water waves. Depending
on the strength of underlying uniform current, we obtain that some particle
trajectories are undulating path to the right or to the left, some are looping
curves with a drift to the right and others are parabolic curves or curves
which have only one loop
Steady water waves with multiple critical layers: interior dynamics
We study small-amplitude steady water waves with multiple critical layers.
Those are rotational two-dimensional gravity-waves propagating over a perfect
fluid of finite depth. It is found that arbitrarily many critical layers with
cat's-eye vortices are possible, with different structure at different levels
within the fluid. The corresponding vorticity depends linearly on the stream
function.Comment: 14 pages, 3 figures. As accepted for publication in J. Math. Fluid
Mec
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