15,408 research outputs found
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
-Strands and Peakon Collisions on
A -strand is a map for a Lie group
that follows from Hamilton's principle for a certain class of -invariant
Lagrangians. Some -strands on finite-dimensional groups satisfy 1+1
space-time evolutionary equations that admit soliton solutions as completely
integrable Hamiltonian systems. For example, the -strand equations
may be regarded physically as integrable dynamics for solitons on a continuous
spin chain. Previous work has shown that -strands for diffeomorphisms on the
real line possess solutions with singular support (e.g. peakons). This paper
studies collisions of such singular solutions of -strands when is the group of diffeomorphisms of the real line
, for which the group product is composition of smooth invertible
functions. In the case of peakon-antipeakon collisions, the solution reduces to
solving either Laplace's equation or the wave equation (depending on a sign in
the Lagrangian) and is written in terms of their solutions. We also consider
the complexified systems of -strand equations for
corresponding to a harmonic map and
find explicit expressions for its peakon-antipeakon solutions, as well.Comment: arXiv:1109.4421 introduced singular solutions of G-strand equations
on the diffeos. This paper solves the equations for their pairwise
interactio
Euler-Poincar\'e equations for -Strands
The -strand equations for a map into a Lie
group are associated to a -invariant Lagrangian. The Lie group manifold
is also the configuration space for the Lagrangian. The -strand itself is
the map , where and are the
independent variables of the -strand equations. The Euler-Poincar\'e
reduction of the variational principle leads to a formulation where the
dependent variables of the -strand equations take values in the
corresponding Lie algebra and its co-algebra,
with respect to the pairing provided by the variational derivatives of the
Lagrangian.
We review examples of different -strand constructions, including matrix
Lie groups and diffeomorphism group. In some cases the -strand equations are
completely integrable 1+1 Hamiltonian systems that admit soliton solutions.Comment: To appear in Conference Proceedings for Physics and Mathematics of
Nonlinear Phenomena, 22 - 29 June 2013, Gallipoli (Italy)
http://pmnp2013.dmf.unisalento.it/talks.shtml, 9 pages, no figure
Factorization of quantum charge transport for non-interacting fermions
We show that the statistics of the charge transfer of non-interacting
fermions through a two-lead contact is generalized binomial, at any temperature
and for any form of the scattering matrix: an arbitrary charge-transfer process
can be decomposed into independent single-particle events. This result
generalizes previous studies of adiabatic pumping at zero temperature and of
transport induced by bias voltage.Comment: 13 pages, 3 figures, typos corrected, references adde
Hamiltonian models for the propagation of irrotational surface gravity waves over a variable bottom
A single incompressible, inviscid, irrotational fluid medium bounded by a
free surface and varying bottom is considered. The Hamiltonian of the system is
expressed in terms of the so-called Dirichlet-Neumann operators. The equations
for the surface waves are presented in Hamiltonian form. Specific scaling of
the variables is selected which leads to approximations of Boussinesq and KdV
types taking into account the effect of the slowly varying bottom. The arising
KdV equation with variable coefficients is studied numerically when the initial
condition is in the form of the one soliton solution for the initial depth.Comment: 18 pages, 6 figures, 1 tabl
On the persistence properties of the cross-coupled Camassa-Holm system
In this paper we examine the evolution of solutions, that initially have
compact support, of a recently-derived system of cross-coupled Camassa-Holm
equations. The analytical methods which we employ provide a full picture for
the persistence of compact support for the momenta. For solutions of the system
itself, the answer is more convoluted, and we determine when the compactness of
the support is lost, replaced instead by an exponential decay rate.Comment: 13 pages, 1 figur
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