342 research outputs found
An integrable shallow water equation with peaked solitons
We derive a new completely integrable dispersive shallow water equation that
is biHamiltonian and thus possesses an infinite number of conservation laws in
involution. The equation is obtained by using an asymptotic expansion directly
in the Hamiltonian for Euler's equations in the shallow water regime. The
soliton solution for this equation has a limiting form that has a discontinuity
in the first derivative at its peak.Comment: LaTeX file. Figure available from authors upon reques
Parametric Representation for the Multisoliton Solution of the Camassa-Holm Equation
The parametric representation is given to the multisoliton solution of the
Camassa-Holm equation. It has a simple structure expressed in terms of
determinants. The proof of the solution is carried out by an elementary theory
of determinanats. The large time asymptotic of the solution is derived with the
fomula for the phase shift. The latter reveals a new feature when compared with
the one for the typical soliton solutions. The peakon limit of the phase shift
ia also considered, showing that it reproduces the known result.Comment: 14 page
Multipeakons and a theorem of Stieltjes
A closed form of the multi-peakon solutions of the Camassa-Holm equation is
found using a theorem of Stieltjes on continued fractions. An explicit formula
is obtained for the scattering shifts.Comment: 6 page
Averaged Template Matching Equations
By exploiting an analogy with averaging procedures in fluid
dynamics, we present a set of averaged template matching equations.
These equations are analogs of the exact template matching equations
that retain all the geometric properties associated with the diffeomorphismgrou
p, and which are expected to average out small scale features
and so should, as in hydrodynamics, be more computationally efficient
for resolving the larger scale features. Froma geometric point of view,
the new equations may be viewed as coming from a change in norm that
is used to measure the distance between images. The results in this paper
represent first steps in a longer termpro gram: what is here is only
for binary images and an algorithm for numerical computation is not
yet operational. Some suggestions for further steps to develop the results
given in this paper are suggested
Chaotic advection in a Rayleigh-Bénard flow
We consider the problem of transport of a passive tracer in the time-dependent flow corresponding to a Rayleigh number scrR slightly above the scrRt at the onset of the even oscillatory instability for Rayleigh-Bénard convection rolls. By modeling the flow with a stream function, we show how to construct and identify invariant structures in the flow that act as a ‘‘template’’ for the motion of fluid particles, in the absence of molecular diffusivity. This approach and symmetry considerations allow us to write explicit formulas that describe the tracer transport for finite times. In the limit of small amplitude of the oscillation, i.e., when (scrR-scrRt)1/2 is small, we show that the amount of fluid transported across a roll boundary grows linearly with the amplitude, in agreement with the experimental and numerical findings of Solomon and Gollub [Phys. Rev. A 38, 6280 (1988)]. The presence of molecular diffusivity introduces a (long) time scale into the problem. We discuss the applicability of the theory in this situation, by introducing a simple rule for determining when the effects of diffusivity are negligible, and perform numerical simulations of the flow in this case to provide an example
The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and dym type
An extension of the algebraic-geometric method for nonlinear integrable PDE's
is shown to lead to new piecewise smooth weak solutions of a class of
-component systems of nonlinear evolution equations. This class includes,
among others, equations from the Dym and shallow water equation hierarchies.
The main goal of the paper is to give explicit theta-functional solutions of
these nonlinear PDE's, which are associated to nonlinear subvarieties of
hyperelliptic Jacobians. The main results of the present paper are twofold.
First, we exhibit some of the special features of integrable PDE's that admit
piecewise smooth weak solutions, which make them different from equations whose
solutions are globally meromorphic, such as the KdV equation. Second, we blend
the techniques of algebraic geometry and weak solutions of PDE's to gain
further insight into, and explicit formulas for, piecewise-smooth finite-gap
solutions.Comment: 31 pages, no figures, to appear in Commun. Math. Phy
Long time behaviour for a class of low-regularity solutions of the Camassa-Holm equation
In this paper, we investigate the long time behaviour for a class of
low-regularity solutions of the Camassa-Holm equation given by the
superposition of infinitely many interacting traveling waves with corners at
their peaks.Comment: 30 page
How to add a boundary condition
Given a conformal QFT local net of von Neumann algebras B_2 on the
two-dimensional Minkowski spacetime with irreducible subnet A\otimes\A, where A
is a completely rational net on the left/right light-ray, we show how to
consistently add a boundary to B_2: we provide a procedure to construct a
Boundary CFT net B of von Neumann algebras on the half-plane x>0, associated
with A, and locally isomorphic to B_2. All such locally isomorphic Boundary CFT
nets arise in this way. There are only finitely many locally isomorphic
Boundary CFT nets and we get them all together. In essence, we show how to
directly redefine the C* representation of the restriction of B_2 to the
half-plane by means of subfactors and local conformal nets of von Neumann
algebras on S^1.Comment: 20 page
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