212 research outputs found
Investigating overtaking collisions of solitary waves in the Schamel equation
This article presents a numerical investigation of overtaking collisions
between two solitary waves in the context of the Schamel equation. Our study
reveals different regimes characterized by the behavior of the wave
interactions. In certain regimes, the collisions maintain two well-separated
crests consistently over time, while in other regimes, the number of local
maxima undergoes variations following the patterns of or .
These findings demonstrate that the geometric Lax-categorization observed in
the Korteweg-de Vries equation (KdV) for two-soliton collisions remains
applicable to the Schamel equation. However, in contrast to the KdV, we
demonstrate that an algebraic Lax-categorization based on the ratio of the
initial solitary wave amplitudes is not feasible for the Schamel equation.
Additionally, we show that the statistical moments for two-solitary wave
collisions are qualitatively similar to the KdV equation and the phase shifts
after soliton interactions are close to ones in integrable KdV and modified KdV
models
Steady transcritical flow over an obstacle: Parametric map of solutions of the forced extended Korteweg-de Vries equation
Transcritical flow of a stratified fluid over an obstacle is often modeled by the forced Korteweg-de Vries equation, which describes a balance among weak nonlinearity, weak dispersion, and small forcing effects. However, in some special circumstances, it is necessary to add an additional cubic nonlinear term, so that the relevant model is the forced extended Korteweg-de Vries equation. Here we seek steady solutions with constant, but different amplitudes upstream and downstream of the forcing region. Our main interest is in the case when the forcing has negative polarity, which represents a hole. The effects of the width of the hole and the amplitude of the hole on these steady solutions are then investigated. © 2011 American Institute of Physics.published_or_final_versio
Interactions of solitons with an external force field: Exploring the Schamel equation framework
This study aims to investigate the interactions of solitons with an external
force within the framework of the Schamel equation, both asymptotically and
numerically. By utilizing asymptotic expansions, we demonstrate that the
soliton interaction can be approximated by a dynamical system that involves the
soliton amplitude and its crest position. To solve the Schamel equation, we
employ a pseudospectral method and compare the obtained results with those
predicted by the asymptotic theory. Remarkably, our findings reveal a
qualitatively agreement between the predictions and the numerical simulations
at early times. Specifically, we classify the soliton interaction into three
categories: (i) steady interaction occurs when the crest of the soliton and the
crest of the external force are in phase, (ii) oscillatory behavior arises when
the soliton's speed and the external force speed are close to resonance,
causing the soliton to bounce back and forth near its initial position, and
(iii) non-reversible motion occurs when the soliton moves away from its initial
position without changing its direction
Nonlinear Lattice Dynamics of Bose-Einstein Condensates
The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine
thermalization in non-metallic solids and develop ``experimental'' techniques
for studying nonlinear problems, continues to yield a wealth of results in the
theory and applications of nonlinear Hamiltonian systems with many degrees of
freedom. Inspired by the studies of this seminal model, solitary-wave dynamics
in lattice dynamical systems have proven vitally important in a diverse range
of physical problems--including energy relaxation in solids, denaturation of
the DNA double strand, self-trapping of light in arrays of optical waveguides,
and Bose-Einstein condensates (BECs) in optical lattices. BECS, in particular,
due to their widely ranging and easily manipulated dynamical apparatuses--with
one to three spatial dimensions, positive-to-negative tuning of the
nonlinearity, one to multiple components, and numerous experimentally
accessible external trapping potentials--provide one of the most fertile
grounds for the analysis of solitary waves and their interactions. In this
paper, we review recent research on BECs in the presence of deep periodic
potentials, which can be reduced to nonlinear chains in appropriate
circumstances. These reductions, in turn, exhibit many of the remarkable
nonlinear structures (including solitons, intrinsic localized modes, and
vortices) that lie at the heart of the nonlinear science research seeded by the
FPU paradigm.Comment: 10 pages, revtex, two-columns, 3 figs, accepted fpr publication in
Chaos's focus issue on the 50th anniversary of the publication of the
Fermi-Pasta-Ulam problem; minor clarifications (and a couple corrected typos)
from previous versio
Long nonlinear internal waves
Author Posting. © Annual Reviews, 2006. This article is posted here by permission of Annual Reviews for personal use, not for redistribution. The definitive version was published in Annual Review of Fluid Mechanics 38 (2006): 395-425, doi:10.1146/annurev.fluid.38.050304.092129.Over the past four decades, the combination of in situ and remote sensing observations has demonstrated that long nonlinear internal solitary-like waves are ubiquitous features of coastal oceans. The following provides an overview of the properties of steady internal solitary waves and the transient processes of wave generation and evolution, primarily from the point of view of weakly nonlinear theory, of which the Korteweg-de Vries equation is the most frequently used example. However, the oceanographically important processes of wave instability and breaking, generally inaccessible with these models, are also discussed. Furthermore, observations often show strongly nonlinear waves whose properties can only be explained with fully nonlinear models.KRH acknowledges
support from NSF and ONR and an Independent Study Award from the
Woods Hole Oceanographic Institution. WKM acknowledges support from NSF and
ONR, which has made his work in this area possible, in close collaboration with former
graduate students at Scripps Institution of Oceanography and MIT
Solitary wave interactions with a periodic forcing: the extended Korteweg-de Vries framework
The aim of this work is to study numerically the interaction of large
amplitude solitary waves with an external periodic forcing using the forced
extended Korteweg-de Vries equation (feKdV). Regarding these interactions, we
find that a solitary wave can bounce back and forth remaining close to its
initial position when the forcing and the solitary wave are near resonant or it
can move away from its initial position without reversing their direction.
Additionally, we verify that the numerical results agree well within the
asymptotic approximation for broad the forcings
Soliton interactions with an external forcing: the modified Korteweg-de Vries framework
The aim of this work is to study asymptotically and numerically the
interaction of solitons with an external forcing with variable speed using the
forced modified Korteweg-de Vries equation (mKdV). We show that the asymptotic
predictions agree well with numerical solutions for forcing with constant speed
and linear variable speed. Regarding forcing with linear variable speed, we
find regimes in which the solitons are trapped at the external forcing and its
amplitude increases or decreases in time depending on whether the forcing
accelerates or decelerates
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