133 research outputs found
Towards the theory of integrable hyperbolic equations of third order
The examples are considered of integrable hyperbolic equations of third order
with two independent variables. In particular, an equation is found which
admits as evolutionary symmetries the Krichever--Novikov equation and the
modified Landau--Lifshitz system. The problem of choice of dynamical variables
for the hyperbolic equations is discussed.Comment: 22
The inverse spectral problem for the discrete cubic string
Given a measure on the real line or a finite interval, the "cubic string"
is the third order ODE where is a spectral parameter. If
equipped with Dirichlet-like boundary conditions this is a nonselfadjoint
boundary value problem which has recently been shown to have a connection to
the Degasperis-Procesi nonlinear water wave equation. In this paper we study
the spectral and inverse spectral problem for the case of Neumann-like boundary
conditions which appear in a high-frequency limit of the Degasperis--Procesi
equation. We solve the spectral and inverse spectral problem for the case of
being a finite positive discrete measure. In particular, explicit
determinantal formulas for the measure are given. These formulas generalize
Stieltjes' formulas used by Krein in his study of the corresponding second
order ODE .Comment: 24 pages. LaTeX + iopart, xypic, amsthm. To appear in Inverse
Problems (http://www.iop.org/EJ/journal/IP
Transfer and scattering of wave packets by a nonlinear trap
In the framework of a one-dimensional model with a tightly localized
self-attractive nonlinearity, we study the formation and transfer (dragging) of
a trapped mode by "nonlinear tweezers", as well as the scattering of coherent
linear wave packets on the stationary localized nonlinearity. The use of the
nonlinear trap for the dragging allows one to pick up and transfer the relevant
structures without grabbing surrounding "garbage". A stability border for the
dragged modes is identified by means of of analytical estimates and systematic
simulations. In the framework of the scattering problem, the shares of trapped,
reflected, and transmitted wave fields are found. Quasi-Airy stationary modes
with a divergent norm, that may be dragged by the nonlinear trap moving at a
constant acceleration, are briefly considered too.Comment: Phys. Rev. E in pres
One-Two Dimensional Nonlinear Pulse Interaction
The peculiar intergrability of the Davey-Stewartson equation allows us to
find analytically solutions describing the simultaneous formation and
interaction of one-dimensional and two-dimensional localized coherent
structures. The predicted phenomenology allows us to address the issue of
interaction of solitons of different dimensionality that may serve as a
starting point for the understanding of hybrido-dimensional collisions recently
observed in nonlinear optical media.Comment: 11 pages + 4 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
Inverse problems associated with integrable equations of Camassa-Holm type; explicit formulas on the real axis, I
The inverse problem which arises in the Camassa--Holm equation is revisited
for the class of discrete densities. The method of solution relies on the use
of orthogonal polynomials. The explicit formulas are obtained directly from the
analysis on the real axis without any additional transformation to a "string"
type boundary value problem known from prior works
Dressing chain for the acoustic spectral problem
The iterations are studied of the Darboux transformation for the generalized
Schroedinger operator. The applications to the Dym and Camassa-Holm equations
are considered.Comment: 16 pages, 6 eps figure
On the tau-functions of the Degasperis-Procesi equation
The DP equation is investigated from the point of view of
determinant-pfaffian identities. The reciprocal link between the
Degasperis-Procesi (DP) equation and the pseudo 3-reduction of the
two-dimensional Toda system is used to construct the N-soliton solution of the
DP equation. The N-soliton solution of the DP equation is presented in the form
of pfaffian through a hodograph (reciprocal) transformation. The bilinear
equations, the identities between determinants and pfaffians, and the
-functions of the DP equation are obtained from the pseudo 3-reduction of
the two-dimensional Toda system.Comment: 27 pages, 4 figures, Journal of Physics A: Mathematical and
Theoretical, to be publishe
Analytic solutions and Singularity formation for the Peakon b--Family equations
Using the Abstract Cauchy-Kowalewski Theorem we prove that the -family
equation admits, locally in time, a unique analytic solution. Moreover, if the
initial data is real analytic and it belongs to with , and the
momentum density does not change sign, we prove that the
solution stays analytic globally in time, for . Using pseudospectral
numerical methods, we study, also, the singularity formation for the -family
equations with the singularity tracking method. This method allows us to follow
the process of the singularity formation in the complex plane as the
singularity approaches the real axis, estimating the rate of decay of the
Fourier spectrum
Self-similarity and singularity formation in a coupled system of Yang-Mills-dilaton evolution equations
We study both analytically and numerically a coupled system of spherically
symmetric SU(2) Yang-Mills-dilaton equation in 3+1 Minkowski space-time. It has
been found that the system admits a hidden scale invariance which becomes
transparent if a special ansatz for the dilaton field is used. This choice
corresponds to transition to a frame rotated in the plane at a
definite angle. We find an infinite countable family of self-similar solutions
which can be parametrized by the - the number of zeros of the relevant
Yang-Mills function. According to the performed linear perturbation analysis,
the lowest solution with N=0 only occurred to be stable. The Cauchy problem has
been solved numerically for a wide range of smooth finite energy initial data.
It has been found that if the initial data exceed some threshold, the resulting
solutions in a compact region shrinking to the origin, attain the lowest N=0
stable self-similar profile, which can pretend to be a global stable attractor
in the Cauchy problem. The solutions live a finite time in a self-similar
regime and then the unbounded growth of the second derivative of the YM
function at the origin indicates a singularity formation, which is in agreement
with the general expectations for the supercritical systems.Comment: 10 pages, 5 figure
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