386 research outputs found
Localization and Coherence in Nonintegrable Systems
We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian
oscillator chains approaching their statistical asympotic states. In systems
constrained by more than one conserved quantity, the partitioning of the
conserved quantities leads naturally to localized and coherent structures. If
the phase space is compact, the final equilibrium state is governed by entropy
maximization and the final coherent structures are stable lumps. In systems
where the phase space is not compact, the coherent structures can be collapses
represented in phase space by a heteroclinic connection to infinity.Comment: 41 pages, 15 figure
Boussinesq/Boussinesq systems for internal waves with a free surface, and the KdV approximation
We study here some asymptotic models for the propagation of internal and
surface waves in a two-fluid system. We focus on the so-called long wave regime
for one dimensional waves, and consider the case of a flat bottom. Starting
from the classical Boussinesq/Boussinesq system, we introduce a new family of
equivalent symmetric hyperbolic systems. We study the well-posedness of such
systems, and the asymptotic convergence of their solutions towards solutions of
the full Euler system. Then, we provide a rigorous justification of the
so-called KdV approximation, stating that any bounded solution of the full
Euler system can be decomposed into four propagating waves, each of them being
well approximated by the solutions of uncoupled Korteweg-de Vries equations.
Our method also applies for models with the rigid lid assumption, and the
precise behavior of the KdV approximations depending on the depth and density
ratios is discussed for both rigid lid and free surface configurations. The
fact that we obtain {\it simultaneously} the four KdV equations allows us to
study extensively the influence of the rigid lid assumption on the evolution of
the interface, and therefore its domain of validity. Finally, solutions of the
Boussinesq/Boussinesq systems and the KdV approximation are rigorously compared
and numerically computed.Comment: To appear in M2A
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
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