783 research outputs found
Hidden solitons in the Zabusky-Kruskal experiment: Analysis using the periodic, inverse scattering transform
Recent numerical work on the Zabusky--Kruskal experiment has revealed,
amongst other things, the existence of hidden solitons in the wave profile.
Here, using Osborne's nonlinear Fourier analysis, which is based on the
periodic, inverse scattering transform, the hidden soliton hypothesis is
corroborated, and the \emph{exact} number of solitons, their amplitudes and
their reference level is computed. Other "less nonlinear" oscillation modes,
which are not solitons, are also found to have nontrivial energy contributions
over certain ranges of the dispersion parameter. In addition, the reference
level is found to be a non-monotone function of the dispersion parameter.
Finally, in the case of large dispersion, we show that the one-term nonlinear
Fourier series yields a very accurate approximate solution in terms of Jacobian
elliptic functions.Comment: 10 pages, 4 figures (9 images); v2: minor revision, version accepted
for publication in Math. Comput. Simula
Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform
The periodic, inverse scattering transform (PIST) is a powerful analytical
tool in the theory of integrable, nonlinear evolution equations. Osborne
pioneered the use of the PIST in the analysis of data form inherently nonlinear
physical processes. In particular, Osborne's so-called nonlinear Fourier
analysis has been successfully used in the study of waves whose dynamics are
(to a good approximation) governed by the Korteweg--de Vries equation. In this
paper, the mathematical details and a new application of the PIST are
discussed. The numerical aspects of and difficulties in obtaining the nonlinear
Fourier (i.e., PIST) spectrum of a physical data set are also addressed. In
particular, an improved bracketing of the "spectral eigenvalues" (i.e., the
+/-1 crossings of the Floquet discriminant) and a new root-finding algorithm
for computing the latter are proposed. Finally, it is shown how the PIST can be
used to gain insightful information about the phenomenon of soliton-induced
acoustic resonances, by computing the nonlinear Fourier spectrum of a data set
from a simulation of internal solitary wave generation and propagation in the
Yellow Sea.Comment: 10 pages, 4 figures (6 images); v2: corrected a few minor mistakes
and typos, version accepted for publication in Math. Comput. Simu
Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
We describe a list of open problems in random matrix theory and the theory of
integrable systems that was presented at the conference Asymptotics in
Integrable Systems, Random Matrices and Random Processes and Universality,
Centre de Recherches Mathematiques, Montreal, June 7-11, 2015. We also describe
progress that has been made on problems in an earlier list presented by the
author on the occasion of his 60th birthday in 2005 (see [Deift P., Contemp.
Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 419-430,
arXiv:0712.0849]).Comment: for Part I see arXiv:0712.084
Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB)
equations, i.e. scalar conservation laws with diffusive-dispersive
regularization. We review the existence of traveling wave solutions for these
two classes of evolution equations. For classical equations the traveling wave
problem (TWP) for a local KdVB equation can be identified with the TWP for a
reaction-diffusion equation. In this article we study this relationship for
these two classes of evolution equations with nonlocal diffusion/dispersion.
This connection is especially useful, if the TW equation is not studied
directly, but the existence of a TWS is proven using one of the evolution
equations instead. Finally, we present three models from fluid dynamics and
discuss the TWP via its link to associated reaction-diffusion equations
Approximated Lax Pairs for the Reduced Order Integration of Nonlinear Evolution Equations
A reduced-order model algorithm, called ALP, is proposed to solve nonlinear
evolution partial differential equations. It is based on approximations of
generalized Lax pairs. Contrary to other reduced-order methods, like Proper
Orthogonal Decomposition, the basis on which the solution is searched for
evolves in time according to a dynamics specific to the problem. It is
therefore well-suited to solving problems with progressive front or wave
propagation. Another difference with other reduced-order methods is that it is
not based on an off-line / on-line strategy. Numerical examples are shown for
the linear advection, KdV and FKPP equations, in one and two dimensions
The Fermi-Pasta-Ulam problem: 50 years of progress
A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with
its suggested resolutions and its relation to other physical problems. We focus
on the ideas and concepts that have become the core of modern nonlinear
mechanics, in their historical perspective. Starting from the first numerical
results of FPU, both theoretical and numerical findings are discussed in close
connection with the problems of ergodicity, integrability, chaos and stability
of motion. New directions related to the Bose-Einstein condensation and quantum
systems of interacting Bose-particles are also considered.Comment: 48 pages, no figures, corrected and accepted for publicatio
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