\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 19, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/19\hfil Solitary wave collisions]
{Solitary wave collisions in the regularized long wave equation}
\author[H. Kalisch, M. H. Y. Nguyen, N. T.Nguyen \hfil EJDE-2013/19\hfilneg]
{Henrik Kalisch, Marie Hai Yen Nguyen, Nguyet Thanh Nguyen} % in alphabetical order
\address{ Henrik Kalisch \newline
Department of Mathematics,
University of Bergen, Postbox 7800, 5020 Bergen, Norway}
\email{henrik.kalisch@math.uib.no}
\address{Marie Hai Yen Nguyen \newline
Laboratoire de m\'et\'eorologie dynamique,
Universit\'{e} Paris 6, 75252 Paris Cedex 05, France}
\email{mathsyen@yahoo.com}
\address{Nguyet Thanh Nguyen \newline
Department of Mathematics,
University of Bergen, Postbox 7800, 5020 Bergen, Norway}
\email{nguyen.nguyet13@yahoo.com}
\thanks{Submitted November 26, 2012. Published January 23, 2013.}
\subjclass[2000]{35Q53, 35B34, 35C08}
\keywords{Solitary waves; solitary-wave interaction; phase shift; resonance}
\begin{abstract}
The regularized long-wave equation admits families of positive and
negative solitary waves. Interactions of these waves are studied,
and it is found that interactions of pairs of positive and pairs of
negative solitary waves feature the same phase shift asymptotically
as the wave velocities grow large as long as the same amplitude ratio is
maintained. The collision of a positive with a negative wave leads to a
host of phenomena, including resonance, annihilation and creation of
secondary waves. A sharp criterion on the resonance for positive-negative
interactions is found.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\allowdisplaybreaks
\section{Introduction}
This article is focused on the interaction
of solitary-wave solutions to the regularized long-wave equation
\begin{equation}\label{equation}
u_t + u_x + (u^2)_x - u_{xxt} = 0,
\end{equation}
which appears as a model equation for surface water waves.
The equation is also known as the BBM or PBBM equation,
as it first appeared in the work of Peregrine \cite{Pe}
and was studied in depth by Benjamin, Bona and Mahoney \cite{BBM}.
The equation was put forward as a model for small amplitude long waves
on the surface of an inviscid incompressible fluid, and as such
is an alternative to the well known Korteweg-de Vries (KdV) equation
\begin{equation}\label{KdV}
u_t + u_x + (u^2)_x + u_{xxx} = 0.
\end{equation}
Both \eqref{equation} and \eqref{KdV} were derived as simplified
models for unidirectional propagation of surface waves,
but the regularized long wave equation has certain advantages,
especially with regard to the numerical
approximation of solutions containing components of shorter wavelength.
Moreover, the linear phase speed of small periodic wave solutions
of \eqref{equation} resembles
the actual phase speed of small amplitude surface waves as described
by the Euler equations more closely than the KdV equation. In particular
the phase velocity of small periodic solutions of \eqref{equation}
is always positive whereas the phase speed can turn negative in the KdV equation.
For a more in-depth explanation of modeling aspects of these equations,
one may consult \cite{AB,BBM,EK,Wh}.
Despite the obvious advantages of \eqref{equation},
the KdV equation \eqref{KdV} has become a generic model for the study
of weakly nonlinear long waves in different types of modeling situations \cite{BOOK},
thanks in part to the completely elastic interaction of its solitary waves \cite{ZK}.
In the context of \eqref{equation} and \eqref{KdV},
solitary-wave solutions may be defined as
progressive waves which propagate without a change in their spatial profile,
which have a single maximum or a single minimum,
and which decay to zero for large absolute values of $x$.
Elastic interaction may be described as follows.
Suppose two solitary waves are arranged initially in such a way
that one wave will pass the other wave (overtaking collision),
or the two waves will meet head-on.
In the KdV equation, which features only overtaking collision,
both waves re-emerge unchanged,
the only remnant of the interaction being a phase shift of both
solitary waves.
The discovery of the elastic interaction of two solitary waves was the first indication
that the KdV equation may represent a completely-integrable,
infinite-dimensional Hamiltonian dynamical system,
and subsequently led to the discovery of an infinite number of
time-invariant integrals \cite{Miura},
and the development of the inverse-scattering method
which can be used to provide exact closed form solutions for a broad class
of initial data \cite{BOOK, GGKM}.
Regarding the equation \eqref{equation},
it was shown in \cite{BPS} that even though the equation has closed form
expressions for exact solitary-wave solutions,
it does not feature elastic interaction of solitary waves.
Indeed, as shown in Figure \ref{couple},
interactions of solitary waves in the model \eqref{equation} generally
lead not only to a phase shift, but also
to the creation of dispersive oscillations which remain after the
interaction.
This finding indicates that the equation \eqref{equation}
is not a completely integrable dynamical system, and in fact,
non-integrability of \eqref{equation} has been proved in \cite{Ol}.
Numerical studies of solitary-wave interactions have been used in a large
number of cases to provide evidence against complete integrability.
A sample of results are studies of the Benjamin and Benjamin-Ono equations
\cite{BoKa,KB}, a higher order compound KdV equation \cite{Kur},
a Boussinesq system for internal waves \cite{ND},
and different types of equations for waves in solids \cite{EST,TS}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{fig1a}
\includegraphics[width=0.49\textwidth]{fig1b}
\end{center}
\caption{\small Interaction of two positive solitary waves in
equation \eqref{equation}. The left panel shows the time evolution
of the spatial profile. The scale is such that the dispersive tail
due to the inelasticity cannot be seen.
The right panel shows the positions of the maxima of the two waves as
functions of $t$. The solid line shows the actual position of the maxima,
while the dashed line indicates the position of the maxima in the case
that no collision has taken place.
}
\label{interact_specialcase}
\end{figure}
While equation \eqref{equation} does not feature an infinite family of conserved
quantities, it does have three independent invariant integrals, which are given by
\begin{align}
\label{conslaws}
I = \int_{- \infty}^{\infty} u \, dx,
\qquad
II = \int_{- \infty}^{\infty} \left( \tfrac{1}{2}u^2
+ \tfrac{1}{2} u_x^2 \right) \, dx,
\quad
III = \int_{- \infty}^{\infty} \left( \tfrac{1}{3} u^3
- \tfrac{1}{2} u_x^2 \right) \, dx.
\end{align}
The solitary-wave solutions
$
u(x,t) = \psi_c(x-ct),
$
of \eqref{equation} are given in terms of the variable $\xi = x-ct$
in the form
\begin{equation}\label{solwav}
\psi_c(\xi) = \frac{3}{2}(c-1) \operatorname{sech}^2 \Big( \tfrac{1}{2}
\sqrt{\tfrac{c-1}{c}} \xi \Big).
\end{equation}
Now as opposed to the situation in the case of the KdV equation which
features only positive solitary waves, the equation \eqref{equation}
admits both positive and negative solitary-wave solutions.
Indeed, it is clear that the expression \eqref{solwav}
actually defines two families of solitary waves.
For the positive solutions, the velocity of the solitary wave is restricted by $c>1$,
and for the negative solutions, the velocity is restricted by $c<0$.
In both cases, the amplitude is given by $A=\frac{3}{2}|(c-1)|$.
Wave profiles for a few positive and negative waves are shown in
Figure \ref{profiles}.
The focus of the present article is two-fold.
In Section 2, we compare the interactions of two positive
and of two negative solitary waves.
After extensive numerical experimentation it appeared
overtaking collisions are classified most effectively by
keeping the amplitude ratio of the two interacting solitary waves constant.
In clear terms, we study the interaction of a solitary wave of
amplitude $A$ with a smaller solitary wave of amplitude $\mathcal{R}A$,
where $\mathcal{R}$ represents the ratio.
If $\mathcal{R}$ is kept constant, then it will be shown in Section 2 that asymptotically as
$A \rightarrow \infty$, the phase shifts of two interacting positive waves
are equal to the phase shifts of two interacting negative waves.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.51\textwidth]{fig2a}
\includegraphics[width=0.47\textwidth]{fig2b}
\end{center}
\caption{\small Positive and negative solitary wave profiles, for
velocities $c=1.1$, $c=2$, $c=3$, and $c=-0.1$, $c=-1$. The right panel
shows a close-up of the waves which shows the different spatial decay
of positive and negative solitary waves.}
\label{profiles}
\end{figure}
The second goal of this paper is the study of head-on collisions
of solitary waves. Here, we investigate a regime in which the
solitary waves are changed dramatically during the collision,
as a considerable part of the available
energy is fed into the nascent secondary solitary waves emerging
after the interaction.
This phenomenon was first discovered by Santarelli \cite{Sa},
and studied in depth by Courtenay Lewis and Tjon \cite{Tjon},
who found that the occurrence of these secondary waves can be
quantified in some sense using a resonance criterion based
on the evaluation of the conserved integrals. However, the
authors of \cite{Tjon} only found an asymptotic characterization
of the resonance,
and it is the purpose of the present work to show numerical
evidence pointing to a sharp resonance criterion.
The numerical method to be used here is a Fourier-collocation method,
where the nonlinear term is treated pseudo-spectrally.
Even though this choice is standard, we recall it briefly in the appendix.
The spectral method is coupled with an explicit four-stage
Runge-Kutta scheme, and the resulting fully discrete code is highly stable
and accurate.
Indeed, it can be shown that the eigenvalues
of the discrete linear operator fall squarely into the domain of $A$-stability
of the Runge-Kutta method. Moreover, spectral convergence in the spatial
discretization is observed, and indeed exponential convergence holds since the
solitary waves used to test the convergence are analytic functions
\cite{BjKa,Ka1,NK}.
It should be mentioned that many other numerical methods for
numerical approximation of solutions of \eqref{equation} have been developed,
and this is still an active area of research.
Recent work featured both Galerkin methods \cite{Om},
finite-difference methods \cite{IsMo}, and collocation methods based on splines
\cite{RaHa}.
A pseudo-spectral method coupled with a leapfrog method for the time-discretization
was proposed in \cite{Slo}, and methods for the study of solitary-wave evolution
can be found in \cite{AlDu}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{fig3a}
\includegraphics[width=0.49\textwidth]{fig3b}
\end{center}
\caption{\small Interaction of two negative solitary waves.
In the right panel, the phase shift and the production of
dispersive oscillations behind the smaller solitary wave
is clearly visible.}
\label{couple}
\end{figure}
\section{Overtaking collisions}
The goal of this section is the comparison of overtaking collisions of two
positive and of two negative solitary waves.
For the comparison of overtaking collisions of a pair of positive
and a pair of negative solitary waves, it appears most convenient
to require a constant ratio between the solitary-wave amplitudes, so that
the parameter space may be defined by the single quantity $A =\frac{3}{2}|(c-1)|$
which represents the amplitude of the larger wave.
Such an approach has actually been advocated in \cite{BySm}, where it was shown
that the change in amplitude of the solitary waves after the interaction
is dependent on the ratio of the amplitudes of the initial solitary waves.
Amplitude changes after interactions have been investigated for positive solitary waves
of \eqref{equation} and for higher-order regularized equations \cite{BPS,Tjon,Mar},
and it was noted in several previous works, that the change
in amplitude is so slight that one might argue that the
identity of the solitary waves is preserved, and it still makes sense
to compute the phase shift of the waves.
Figure \ref{3to2} shows the result of several runs with different amplitudes.
If seen in relation to the direction of propagation, in both cases, the
larger wave experiences a forward shift, while the smaller wave
experiences a backward shift.
In the left panel of Figure \ref{3to2}, the forward shift of the larger
solitary wave after the interaction is plotted for both the positive
and the negative wave.
The amplitude ratio is fixed at $3:2$ in Figure \ref{3to2},
so that $\mathcal{R}=\tfrac{2}{3}$,
both for the interaction of two positive waves and for
the interaction of two negative waves.
The data from the numerical runs for two positive waves are shown in the figures circles,
and data for two negative waves are shown as dots.
A rational curvefit is used for both the forward phase shift $\theta_L$
of the larger wave, and for the backwards shift $\theta_S$
of the smaller wave. The curve fit uses the simple model
$$
|\theta_L| = \frac{P_1A + P_2}{A + Q_1}
\quad \mbox{ and } \quad
|\theta_S| = \frac{p_1A + p_2}{A + q_1}.
$$
The resulting horizontal asymptotes $P_1$ and $p_1$ are plotted as dashed lines.
There is no visually discernible difference between the asymptotes,
and the two asymptotic values $P_1$ and $p_1$ also lie within each other's
confidence intervals for the curvefit.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{fig4a}
\includegraphics[width=0.49\textwidth]{fig4b}
\end{center}
\caption{\small Overtaking collisions of a large and a small solitary wave with a constant
amplitude ratio of $3:2$. The left panel shows the magnitude of the phase shift
of the larger waves $|\theta_L|$, and the right panel shows the magnitude of the
phase shift of the smaller waves $|\theta_S|$.
The circles denote the interaction of two positive waves, and
the dots denote the interaction of two negative waves.
The solid curves represent a rational curve fit, and the dashed line is the
horizontal asymptote of both curve fits.}
\label{3to2}
\end{figure}
The results of a similar study for a constant amplitude ratio $3:1$ are
shown in Figure \ref{3to1}, and further test cases have been run for various other
amplitude ratios. The results of these studies are all indicative of the basic relation
that the phase shift in the small and large solitary wave are asymptotically
equal after the interaction of a pair of positive
and the interaction of a pair of negative solitary waves as long as the amplitude
ratio $\mathcal{R}$ between the larger and the smaller wave is kept constant.
Note that the magnitude of the phase shifts of two positive waves
becomes very large for small amplitudes, while the
phase shifts of two negative waves becomes rather smaller.
If it is assumed that the phase shift is in some sense proportional
to the interaction time, then the reason for this difference may be found
in the different profiles of the positive and negative waves.
Indeed, the positive waves become wider with decreasing
amplitude, while the negative waves become narrower with decreasing amplitude
(cf. Figure \ref{profiles}).
Thus, in view of the relatively heavier tails,
the interaction time
is comparatively longer for two small positive solitary waves
than it is for two small negative solitary waves,
even though the velocities of the negative waves are smaller.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{fig5a}
\includegraphics[width=0.49\textwidth]{fig5b}
\end{center}
\caption{\small Overtaking collisions of a large and a small solitary wave with a constant
amplitude ratio of $3:1$. The left panel shows the magnitude of the phase shift
of the larger waves $|\theta_L|$, and the right panel shows the magnitude of the
phase shift of the smaller waves $|\theta_S|$.
The circles denote the interaction of two positive waves, and
the dots denote the interaction of two negative waves.
The solid curves represent a rational curve fit, and the dashed line is the
horizontal asymptote of both curve fits.}
\label{3to1}
\end{figure}
\section{Head on collisions}
The inelasticity of the regularized long wave equation manifests itself
somewhat differently in the case of two waves of opposite sign.
While the interaction of two solitary waves of the same polarity
produces only a dispersive tail, the collision of a positive and negative
solitary wave can lead to the creation of secondary solitary waves
in addition to a dispersive tail. It is also possible for two solitary
waves of opposite polarity to be annihilated by the interaction.
The precise nature of a head-on collision depends
on the two waves being close to resonance, and the
outcome of the interaction near resonance may be characterized as follows.
If both solitary waves are of small amplitude, then annihilation takes place.
In other words, the only remaining disturbance after the interaction is
a dispersive tail (see Fig. 5 in \cite{Tjon}). For larger amplitudes, the
waves re-emerge out of the dispersive tail,
and for even larger amplitudes, secondary solitary waves
appear after the interaction.
A typical case of a resonant interaction of two large amplitude
wave of opposite polarity is shown in Figure \ref{Resonance}.
In the following, a numerical study of head-on collisions is presented,
and a sharp resonance criterion for the interaction of a positive and a negative
solitary wave is exhibited. This result is an improvement upon the work
of Courtenay Lewis and Tjon \cite{Tjon}, who investigated the resonance
which was originally discovered by Santarelli \cite{Sa}.
Denoting the positive solitary wave by $\psi_{c_p}$, and the negative
solitary wave by $\psi_{c_n}$, the resonance criterion was given
by Courtenay Lewis and Tjon \cite{Tjon}
in terms of
$I_n = \int \psi_{c_n}$ and $I_p= \int \psi_{c_p}$
by $I_p + I_n= 0$. Indeed, using
$$
r = \frac{I_p + I_n}{I_p - I_n}
$$
to parameterize the trial space, they found the resonance
near but not on the line $r=0$.
Moreover, it was found that as the total area
$I_p - I_n$ increases, the resonance moved closer to the
line $I_p + I_n$ = 0, which can be written
in terms of the phase velocities as
\begin{equation}\label{tjon}
c_p + c_n = 1.
\end{equation}
While the use of $I$ to parameterize the trial space may appear natural
from the viewpoint of completely integrable differential equations,
viewing the solution set as parameterized by the wave speed $c$
is more useful for pinpointing the exact resonance condition.
As will be clear from the numerical experiments presented here,
the resonance can be characterized explicitly by the condition
\begin{equation}\label{KNN}
c_p + c_n = 0.85.
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig6}
\end{center}
\caption{\small Interaction of a positive and a negative solitary wave at resonance.
The wave speed of the positive solitary wave is $c=10$.
The wave speed of the negative solitary wave is $c=-9.15$.
The figure shows snapshots of the solutions at different times, as indicated
at the right end of the respective curve.
A violent, nearly singular interaction can be seen, and
the formation of secondary solitary waves is observed as the main waves pull from
the interaction region.}
\label{Resonance}
\end{figure}
In order to facilitate comparison with the work in \cite{Tjon},
we choose the same method to quantify the resonance by way of
the invariant integral $II$.
In fact, as noted in \cite{Tjon}, one may use any one of the
three conserved integrals $I$, $II$ $III$, or a linear
combination of these, but the advantage of $II$ is that it is automatically
positive throughout a computation.
Owing to the fast decay of the
exact solitary-wave solutions, one may define these integrals for individual
components of initial data. So if initial data are taken to be
$u_0 = \psi_{c_p}(x) + \psi_{c_n}(x-\tau)$,
then one may define $II_p = II( \psi_{c_p})$ and $II_n = II( \psi_{c_n})$.
The same may be done after an interaction if the waves have separated from
each other, and from any dispersive residue. This leads to
$II_p^f$ and $II_n^f$, where the superscripts indicate that the integrals
are computed at the final time after the interaction.
Now to quantify the resonance, one may use the quantity
$$
\kappa = 1 - \frac{II_p^f + II_n^f}{II_p + II_n}.
$$
Note that $\kappa$ is always between $0$ and $1$, and $\kappa = 0$ signifies
the case where both solitary waves re-emerge unchanged from the interaction.
For the equation \eqref{equation}, $\kappa$ is never exactly zero, because of
the inelasticity. On the other hand, the value $\kappa = 1$ indicates that
the original solitary waves have completely disappeared.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.49\textwidth]{fig7a}
\includegraphics[width=0.49\textwidth]{fig7b}
\end{center}
\caption{\small The left panel displays the inelasticity $\kappa$
of the head-on collision of a positive and a negative solitary wave
as a function of $c_p+c_n$.
The positive solitary wave is kept constant at the speed $c_p=6$,
and the velocity of the negative solitary wave is varied.
The largest value of $\kappa$ appears precisely at $c_p+c_n=0.85$. The right panel
shows data from the same experiments, and records the number of secondary positive
and negative solitary waves created after the collision. The number of positive secondary
waves is graphed with dots, and the number of negative waves is graphed with an $\times$.}
\label{c1lusc2is6}
\end{figure}
Figure \ref{c1lusc2is6} shows the result of a number of numerical runs
for positive-negative solitary-wave interactions. In the left panel,
the resonance parameter $\kappa$ is graphed against the sum of the
velocities $c_p + c_n$. It is plain from the figure that the largest
value of $\kappa$ occurs precisely on the line $c_p + c_n = 0.85$.
The right panel of Figure \ref{c1lusc2is6} indicates the number
of secondary solitary waves created by the collision of the original waves,
and it is again clear that the maximal number of secondary waves is achieved
exactly on the line $c_p + c_n = 0.85$.
Figure \ref{c1lusc2is10} displays the results of similar runs but now with
a positive wave of fixed phase speed $c=10$. Note that the number of secondary
waves is higher in these trials, and it appears that by choosing
initial solitary waves of large enough speed, one may create any number
of secondary waves.
\begin{figure}
\begin{center}
\includegraphics[width=0.49\textwidth]{fig8a}
\includegraphics[width=0.49\textwidth]{fig8b}
\end{center}
\caption{\small The left panel displays the inelasticity $\kappa$
of the head-on collision of a positive and a negative solitary wave
as a function of $c_p+c_n$.
The positive solitary wave is kept constant at the speed $c_p=10$,
and the velocity of the negative solitary wave is varied.
The largest value of $\kappa$ appears precisely at $c_p+c_n=0.85$. The right panel
shows data from the same experiments, and records the number of secondary positive
and negative solitary waves created after the collision. The number of positive secondary
waves is graphed with dots, and the number of negative waves is graphed with an $\times$.}
\label{c1lusc2is10}
\end{figure}
\section{Conclusions}
In this paper, two aspects of solitary-wave interactions were
investigated. First, the overtaking collision of a pair of positive
and a pair of negative solitary waves was compared, and it was
shown numerically that the phase shift of the waves is asymptotically
equal if the amplitude ratio of the waves is held constant. The approach is
in line with previous work \cite{BySm,Mar} which suggested the
amplitude ratio as a convenient measure for properties of solitary-wave
interactions.
Secondly, the head-on collision of a pair of solitary waves of opposite
polarity was studied. It was shown that the resonance parameter $\kappa$
is a convenient measure for the behavior of the solution, and that
resonance occurs precisely on the line $c_p + c_n = 0.85$.
This resonance condition is an improvement upon previously available results.
At and near resonance, creation of secondary solitary waves is observed.
Annihilation is observed when the amplitudes of the solitary waves are
sufficiently small.
While the equation \eqref{equation}
is known to be a reasonable model for long waves of small amplitude
and negligible transverse variation,
it is also apparent that most of the waves shown in this paper have large amplitude, so
that they do not lie within the regime of physical applicability
of equation \eqref{equation} as a long wave model for surface water
waves. However, since \eqref{equation} is one of the key models
for water waves,
it is also important to have a solid understanding of the dynamics
of solutions from a mathematical point of view.
In the case of the KdV equation \eqref{KdV}, the inverse-scattering
theory \cite{BOOK} provides a convenient framework of the mathematical
study of the equations. In the case of \eqref{equation}, these methods
are not available, and therefore the analysis of mathematical properties
is more difficult. Nevertheless, a number of rigorous results exists,
such as proofs of well posedness \cite{BBM,Ch1},
and studies investigating the relation between the
periodic and pure Cauchy problem \cite{Ch2,Pa}.
Some recent work focuses on establishing precise estimates
on the change in amplitude and the phase shift in the overtaking collision
of two positive solitary waves \cite{MM1,MMM},
but it is unclear whether these techniques will also apply to solitary-wave
interactions featuring one or two negative solitary waves.
The dynamic stability of positive solitary was established
some time ago \cite{Be1,Bo,GSS}, and has also been studied numerically \cite{BMR}.
However, small negative waves may be unstable, as explained in
\cite{Ka2}, and proved in \cite{KN,NK}.
The instability of small negative solitary waves may
be explained by the inability of coherent structures to withstand
the dispersion of the linear part of the equation \cite{Al}.
This phenomenon may also be invoked to explain the annihilation
of a positive and a negative solitary wave in a head-on collision.
One may think of the interaction as conserving the total energy $II(u)$,
and if the two solitary waves are near resonance, then
the energy is fed into secondary waves. However, due to the instability
of small negative solitary waves, the secondary negative wave disperses
immediately.
This also explains why the number of negative secondary waves is generally
smaller than the number of positive secondary waves.
\section{Appendix: The numerical technique}
The spectral projection of the initial-value problem associated
to the evolution equation \eqref{equation} is briefly recalled.
In order to approximate the problem on the real line, a large interval
$[0,L]$ is chosen. The problem is then translated to the interval $[0,2\pi]$
by the scaling $u(ax,t)=v(x,t),$ where $a=\frac{L}{2\pi}.$
The evolution equation satisfied by $v$ is then
\begin{displaymath}
\begin{array}{ll}
a^2 v_t(x,t) + a v_x(x,t) + a (v^2)_x(x,t) - v_{xxt}(x,t) = 0, & x \in [0,2\pi], ~t > 0,\\
\end{array}
\end{displaymath}
and the initial-value problem is obtained by setting periodic boundary conditions
$v(0,t) = v(2\pi,t), t \ge 0$, and initial data
$v(x,0) = u_0 (ax), x \in [0,2\pi]$.
This approach is standard, and may be found in any treatment of spectral methods.
A discussion regarding different aspects of the approximation of the problem on the real line
by a periodic problem may be found in \cite{Ch2} and \cite{Pa}.
Discretizing using a Fourier collocation method yields
\begin{equation}\label{discrete_eq}
\begin{gathered}
\frac{\partial}{\partial t} \hat{v}_N(k,t) = -\frac{aik}{a^2+k^2}
\Big\{ \hat{v}_N(k,t)+\mathcal{F}\Big(\big[\mathcal{F}^{-1}
(\hat{v}_N)\big]^2\Big ) \Big\},\\
k=-\frac{N}{2}+1,\dots ,\frac{N}{2},\; t > 0,
\end{gathered}
\end{equation}
where
$\mathcal{F}$ is the discrete Fourier transform defined for an arbitrary continuous
function $w$ by $\mathcal{F} w(k) = \frac{1}{N} \sum_{j=0}^{N-1} e^{-ikx_j} w(x_j)$.
The symbol $\mathcal{F}^{-1}$ denotes the discrete
inverse Fourier transform, given by
$\mathcal{F}^{-1}(\hat{w},x_j) = \sum_{k=-\frac{N}{2}+1}^\frac{N}{2} e^{ikx_j}\hat{w}(k,t)$,
evaluated at the collocation points
$x_j=\frac{2\pi j}{N},$ for $j=1,\dots ,N$.
The system \eqref{discrete_eq}
is a system of $N$ ordinary differential equations for the discrete Fourier
coefficients $\hat{v}_N(k,t),$ for $k = -\frac{N}{2}+1,\dots ,\frac{N}{2}.$
As is customary, the coefficient $\hat{v}_N(\frac{N}{2},t)$ is set to zero,
but carried along for the discrete Fourier transform.
We integrate the system by using a four-stage explicit Runge-Kutta scheme
with a uniform time step $h$.
To test the convergence of the algorithm and the numerical
implementation, the normalized discrete $L^2$-norm is used.
This norm is defined by
$$
\|v(\cdot,t)\|_{N,2}^2 = \frac{1}{N} \sum_{i=1}^N |v(x_i,t)|^2.
$$
The relative $L^2$-error is then defined to be
$$
\mbox{Error}=\frac{\|v-v_N\|_{N,2}}{\|v\|_{N,2}},
$$
where $v(x_i,t)$ is the exact solution, and $v_N(x_i,t)$ is the numerical
approximation at a specific time $t$.
In order to test the implementation of the algorithm, the evolution of
a solitary-wave solution is computed numerically,
and then compared to the exact solutions obtained by translating the wave by
an appropriate distance.
Table \ref{combinations} displays the outcome of several runs with varying
number of modes $N$ and time step $h$. It is clear that both the required
$4$-th order convergence in terms of the time step $h$
and the spectral convergence in terms of the number of spatial grid points $N$
is achieved.
%
\begin{table}[h]
$$
\begin{tabular}{ccclccr}
\hline
\multicolumn{3}{c}{Temporal discretization} & & \multicolumn{3}{c}{Spatial discretization}\\
\hline
h & Error & ratio & \qquad \qquad &
N & Error & ratio \\
\hline
0.1000 & 7.8226e-05 & & & 1024 & 4.921e-01 & \\
0.0500 & 4.4138e-06 & 17.723 & & 2048 & 2.378e-01 & 2.07 \\
0.0250 & 2.6056e-07 & 16.940 & & 4096 & 2.125e-02 & 11.19 \\
0.0125 & 1.5801e-08 & 16.490 & & 8192 & 1.968e-04 & 107.69 \\
0.0063 & 9.7229e-10 & 16.251 & & 16384 & 2.431e-08 & 8097.02 \\
0.0031 & 6.0236e-11 & 16.142 & & 32768 & 1.335e-09 & 1.82 \\
0.0016 & 3.7116e-12 & 16.230 & & & & \\
0.0008 & 2.1690e-13 & 17.112 & & & & \\
\hline
\end{tabular}
$$
\caption{\small Discretization errors arising on a domain $[0,200]$, at the final time $T=8$.
The first three columns show errors achieved with a fixed number of grid points $N=4096$.
The last three columns show errors achieved with a fixed time step of $h=0.001$.}
\label{combinations}
\end{table}
\subsection*{Acknowledgements}
This research was supported in part by the
Research Council of Norway through grant no. NFR 213474/F20.
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\end{document}