395 research outputs found
An explicit finite difference scheme for the Camassa-Holm equation
We put forward and analyze an explicit finite difference scheme for the
Camassa-Holm shallow water equation that can handle general initial data
and thus peakon-antipeakon interactions. Assuming a specified condition
restricting the time step in terms of the spatial discretization parameter, we
prove that the difference scheme converges strongly in towards a
dissipative weak solution of Camassa-Holm equation.Comment: 45 pages, 6 figure
Some Remarks on the KP System of the Camassa-Holm Hierarchy
We study a Kadomtsev-Petviashvili system for the local Camassa-Holm hierarchy
obtaining a candidate to the Baker-Akhiezer function for its first reduction
generalizing the local Camassa-Holm. We focus our attention on the differences
with the standard KdV-KP case.Comment: This is a contribution to the Proc. of workshop on Geometric Aspects
of Integrable Systems (July 17-19, 2006; Coimbra, Portugal), published in
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
An integrable semi-discretization of the Camassa-Holm equation and its determinant solution
An integrable semi-discretization of the Camassa-Holm equation is presented.
The keys of its construction are bilinear forms and determinant structure of
solutions of the CH equation. Determinant formulas of -soliton solutions of
the continuous and semi-discrete Camassa-Holm equations are presented. Based on
determinant formulas, we can generate multi-soliton, multi-cuspon and
multi-soliton-cuspon solutions. Numerical computations using the integrable
semi-discrete Camassa-Holm equation are performed. It is shown that the
integrable semi-discrete Camassa-Holm equation gives very accurate numerical
results even in the cases of cuspon-cuspon and soliton-cuspon interactions. The
numerical computation for an initial value condition, which is not an exact
solution, is also presented
Integration of the EPDiff equation by particle methods
The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold
A dressing method for soliton solutions of the Camassa-Holm equation
The soliton solutions of the Camassa-Holm equation are derived by the
implementation of the dressing method. The form of the one and two soliton
solutions coincides with the form obtained by other methods.Comment: 18 pages, 2 figure
Discrete Variational Derivative Methods for the EPDiff equation
The aim of this paper is the derivation of structure preserving schemes for
the solution of the EPDiff equation, with particular emphasis on the two
dimensional case. We develop three different schemes based on the Discrete
Variational Derivative Method (DVDM) on a rectangular domain discretized with a
regular, structured, orthogonal grid.
We present numerical experiments to support our claims: we investigate the
preservation of energy and linear momenta, the reversibility, and the empirical
convergence of the schemes. The quality of our schemes is finally tested by
simulating the interaction of singular wave fronts.Comment: 41 pages, 41 figure
Decoupled and unidirectional asymptotic models for the propagation of internal waves
We study the relevance of various scalar equations, such as inviscid
Burgers', Korteweg-de Vries (KdV), extended KdV, and higher order equations (of
Camassa-Holm type), as asymptotic models for the propagation of internal waves
in a two-fluid system. These scalar evolution equations may be justified with
two approaches. The first method consists in approximating the flow with two
decoupled, counterpropagating waves, each one satisfying such an equation. One
also recovers homologous equations when focusing on a given direction of
propagation, and seeking unidirectional approximate solutions. This second
justification is more restrictive as for the admissible initial data, but
yields greater accuracy. Additionally, we present several new coupled
asymptotic models: a Green-Naghdi type model, its simplified version in the
so-called Camassa-Holm regime, and a weakly decoupled model. All of the models
are rigorously justified in the sense of consistency
Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation
We study the dynamics of measure-valued solutions of what we call the EPDiff
equations, standing for the {\it Euler-Poincar\'e equations associated with the
diffeomorphism group (of or an -dimensional manifold )}.
Our main focus will be on the case of quadratic Lagrangians; that is, on
geodesic motion on the diffeomorphism group with respect to the right invariant
Sobolev metric. The corresponding Euler-Poincar\'e (EP) equations are the
EPDiff equations, which coincide with the averaged template matching equations
(ATME) from computer vision and agree with the Camassa-Holm (CH) equations in
one dimension. The corresponding equations for the volume preserving
diffeomorphism group are the well-known LAE (Lagrangian averaged Euler)
equations for incompressible fluids. We first show that the EPDiff equations
are generated by a smooth vector field on the diffeomorphism group for
sufficiently smooth solutions. This is analogous to known results for
incompressible fluids--both the Euler equations and the LAE equations--and it
shows that for sufficiently smooth solutions, the equations are well-posed for
short time. In fact, numerical evidence suggests that, as time progresses,
these smooth solutions break up into singular solutions which, at least in one
dimension, exhibit soliton behavior. With regard to these non-smooth solutions,
we study measure-valued solutions that generalize to higher dimensions the
peakon solutions of the (CH) equation in one dimension. One of the main
purposes of this paper is to show that many of the properties of these
measure-valued solutions may be understood through the fact that their solution
ansatz is a momentum map. Some additional geometry is also pointed out, for
example, that this momentum map is one leg of a natural dual pair.Comment: 27 pages, 2 figures, To Alan Weinstein on the occasion of his 60th
Birthda
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