43 research outputs found
NUMERICAL SCHEMES FOR COMPUTING DISCONTINUOUS SOLUTIONS OF THE DEGASPERIS-PROCESI EQUATION
Recent work [4] has shown that the Degasperis-Procesi equation is well-posed in the class of (discontinuous) entropy solutions. In the present paper we construct numerical schemes and prove that they converge to entropy solutions. Additionally, we provide several numerical examples accentuating that discontinuous (shock) solutions form independently of the smoothness of the initial data. Our focus on discontinuous solutions contrasts notably with the existing literature on the Degasperis-Procesi equation, which seems to emphasize similarities with the Camassa-Holm equation (bi-Hamiltonian structure, integrabillity, peakon solutions, H1 as the relevant functional space)
On asymptotically equivalent shallow water wave equations
The integrable 3rd-order Korteweg-de Vries (KdV) equation emerges uniquely at
linear order in the asymptotic expansion for unidirectional shallow water
waves. However, at quadratic order, this asymptotic expansion produces an
entire {\it family} of shallow water wave equations that are asymptotically
equivalent to each other, under a group of nonlinear, nonlocal, normal-form
transformations introduced by Kodama in combination with the application of the
Helmholtz-operator. These Kodama-Helmholtz transformations are used to present
connections between shallow water waves, the integrable 5th-order Korteweg-de
Vries equation, and a generalization of the Camassa-Holm (CH) equation that
contains an additional integrable case. The dispersion relation of the full
water wave problem and any equation in this family agree to 5th order. The
travelling wave solutions of the CH equation are shown to agree to 5th order
with the exact solution
Long-time asymptotics for the Degasperis-Procesi equation on the half-line
We analyze the long-time asymptotics for the Degasperis--Procesi equation on
the half-line. By applying nonlinear steepest descent techniques to an
associated -matrix valued Riemann--Hilbert problem, we find an
explicit formula for the leading order asymptotics of the solution in the
similarity region in terms of the initial and boundary values.Comment: 61 pages, 11 figure
Integrability of Differential-Difference Equations with Discrete Kinks
In this article we discuss a series of models introduced by Barashenkov,
Oxtoby and Pelinovsky to describe some discrete approximations to the \phi^4
theory which preserve travelling kink solutions. We show, by applying the
multiple scale test that they have some integrability properties as they pass
the A_1 and A_2 conditions. However they are not integrable as they fail the
A_3 conditions.Comment: submitted to the Proceedings of the workshop "Nonlinear Physics:
Theory and Experiment.VI" in a special issue di Theoretical and Mathematical
Physic
Degasperis-Procesi peakons and the discrete cubic string
We use an inverse scattering approach to study multi-peakon solutions of the
Degasperis-Procesi (DP) equation, an integrable PDE similar to the Camassa-Holm
shallow water equation. The spectral problem associated to the DP equation is
equivalent under a change of variables to what we call the cubic string
problem, which is a third order non-selfadjoint generalization of the
well-known equation describing the vibrational modes of an inhomogeneous string
attached at its ends. We give two proofs that the eigenvalues of the cubic
string are positive and simple; one using scattering properties of DP peakons,
and another using the Gantmacher-Krein theory of oscillatory kernels.
For the discrete cubic string (analogous to a string consisting of n point
masses) we solve explicitly the inverse spectral problem of reconstructing the
mass distribution from suitable spectral data, and this leads to explicit
formulas for the general n-peakon solution of the DP equation. Central to our
study of the inverse problem is a peculiar type of simultaneous rational
approximation of the two Weyl functions of the cubic string, similar to
classical Pade-Hermite approximation but with lower order of approximation and
an additional symmetry condition instead. The results obtained are intriguing
and nontrivial generalizations of classical facts from the theory of Stieltjes
continued fractions and orthogonal polynomials.Comment: 58 pages, LaTeX with AMS packages, to appear in International
Mathematics Research Paper