19 research outputs found
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
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
Particle paths beneath forced small amplitude periodic waves in a shallow channel with constant vorticity
Particle paths beneath small amplitude periodic forced waves in a shallow water channel are investigated.
The problem is formulated in the forced Korteweg-de Vries equation framework which allows to approximate the velocity field in the bulk fluid. We show that the flow can have zero, one or three stagnation points.
Besides, differently from the unforced problem, stagnation points can arise for small values of the vorticity as long as the moving disturbance travels sufficiently fast
Bipolar solitary wave interactions within the Schamel equation
Pair soliton interactions play a significant role in the dynamics of soliton
turbulence. The interaction of solitons with different polarities is
particularly crucial in the context of abnormally large wave formation, often
referred to as freak or rogue waves, as these interactions result in an
increase in the maximum wave field. In this article, we investigate the
features and properties of bipolar soliton interactions within the framework of
the non-integrable Schamel equation, contrasting them with the integrable
modified Korteweg-de Vries equation. We examine variations in moments and
extrema of the wave fields. Additionally, we identify scenarios in which, in
the bipolar solitary wave interaction, the smaller solitary wave transfers a
portion of its energy to the larger one, causing an increase in the amplitude
of the larger solitary wave and a decrease in the amplitude of the smaller one,
returning them to their pre-interaction state. Notably, we observe that
non-integrability can be considered a factor that triggers the formation of
rogue waves
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
Solitary wave collisions for Whitham-Boussinesq systems
This work concerns soliton-type numerical solutions for two
Whitham-Boussinesq-type models. Solitary waves are computed using an iterative
Newton-type and continuation methods with high accuracy. The method allow us to
compute solitary waves with large amplitude and speed close to the singular
limit. These solitary waves are set as initial data and overtaking collisions
are considered for both systems. We show that both system satisfy the geometric
Lax-categorization of two-soliton collision. Numerical evidences indicate that
one of the systems also admits an algebraic Lax-categorization based on the
ratio of the initial solitary wave amplitudes with a different range from the
one predicted by Lax. However, we show that such categorization is not possible
for the second system
An investigation of the flow structure beneath solitary waves with constant vorticity on a conducting fluid under normal electric fields
The motion of an interface separating two fluids under the effect of electric
fields is a subject that has picked the attention of researchers from different
areas. While there is an abundance of studies investigating the free surface
wave properties, very few works have examined the associated velocity field
within the bulk of the fluid. Therefore, in this paper, we investigate
numerically the flow structure beneath solitary waves with constant vorticity
on an inviscid conducting fluid bounded above by a dielectric gas under normal
electric fields in the framework of a weakly nonlinear theory. Elevation and
depression solitary waves with constant vorticity are computed by a
pseudo-spectral method and a parameter sweep on the intensity of the electric
field is carried out in order to study its role in the appearance of stagnation
points. We find that for elevation solitary waves the location of stagnation
points does not change significantly with variations of the electric field. For
depression solitary waves, on the other hand, the electric field acts as a
catalyser that makes possible the appearance of stagnation points - in the
sense that in its absence there is no stagnation point