620 research outputs found
Generalised Fourier Transform and Perturbations to Soliton Equations
A brief survey of the theory of soliton perturbations is presented. The focus
is on the usefulness of the so-called Generalised Fourier Transform (GFT). This
is a method that involves expansions over the complete basis of `squared
olutions` of the spectral problem, associated to the soliton equation. The
Inverse Scattering Transform for the corresponding hierarchy of soliton
equations can be viewed as a GFT where the expansions of the solutions have
generalised Fourier coefficients given by the scattering data.
The GFT provides a natural setting for the analysis of small perturbations to
an integrable equation: starting from a purely soliton solution one can
`modify` the soliton parameters such as to incorporate the changes caused by
the perturbation.
As illustrative examples the perturbed equations of the KdV hierarchy, in
particular the Ostrovsky equation, followed by the perturbation theory for the
Camassa- Holm hierarchy are presented.Comment: 20 pages, no figures, to appear in: Discrete and Continuous Dynamical
Systems
The scattering approach for the Camassa-Holm equation
We present an approach proving the integrability of the Camassa--Holm
equation for initial data of small amplitude
Camassa-Holm type equations for axisymmetric Poiseuille pipe flows
We present a study on the nonlinear dynamics of a disturbance to the laminar
state in non-rotating axisymmetric Poiseuille pipe flows. The associated
Navier-Stokes equations are reduced to a set of coupled generalized
Camassa-Holm type equations. These support singular inviscid travelling waves
with wedge-type singularities, the so called peakons, which bifurcate from
smooth solitary waves as their celerity increase. In physical space they
correspond to localized toroidal vortices or vortexons. The inviscid vortexon
is similar to the nonlinear neutral structures found by Walton (2011) and it
may be a precursor to puffs and slugs observed at transition, since most likely
it is unstable to non-axisymmetric disturbances.Comment: 11 pages, 4 figures, 31 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Smooth and Peaked Solitons of the CH equation
The relations between smooth and peaked soliton solutions are reviewed for
the Camassa-Holm (CH) shallow water wave equation in one spatial dimension. The
canonical Hamiltonian formulation of the CH equation in action-angle variables
is expressed for solitons by using the scattering data for its associated
isospectral eigenvalue problem, rephrased as a Riemann-Hilbert problem. The
momentum map from the action-angle scattering variables
to the flow momentum () provides the Eulerian representation of
the -soliton solution of CH in terms of the scattering data and squared
eigenfunctions of its isospectral eigenvalue problem. The dispersionless limit
of the CH equation and its resulting peakon solutions are examined by using an
asymptotic expansion in the dispersion parameter. The peakon solutions of the
dispersionless CH equation in one dimension are shown to generalize in higher
dimensions to peakon wave-front solutions of the EPDiff equation whose
associated momentum is supported on smoothly embedded subspaces. The Eulerian
representations of the singular solutions of both CH and EPDiff are given by
the (cotangent-lift) momentum maps arising from the left action of the
diffeomorphisms on smoothly embedded subspaces.Comment: First version -- comments welcome! Submitted to JPhys
- …