3.1. Asymptotic Theory
In this section we introduce the asymptotic theory that allows to turn the study of the partial differential Equation (
2) into a dynamical system. This idea has been used in different contexts [
13,
14]; in particular for the interest reader we mention the recent works of Frassu and Viglialoro [
19] and Frassu et al. [
20], where the authors analyze dynamical systems modeling chemotaxis mechanisms formulated through partial differential equations.
Asymptotic results on the interaction of a solitary wave with an external force were first reported by Grimshaw and Pelinovsky [
17]. For the sake of completeness, we recall their main results assuming that
as
. For a weak external force (
), we seek for a slowly time-varying solitary wave with expansion
where
is the position of the crest of the wave. At first order, the wave profile is given by
In particular, the amplitude variation as a function of time can be obtained from the first-order momentum Equation (
4)
and its rate of change at first-order, which is given by
Notice that
is a function of
, thus the dynamical system (
10)–(12) describes the amplitude and the position of the crest of the solitary wave solution. Assuming a broad external force, the momentum equations reads
Moreover, in the weak-amplitude solitary wave regime (
), the quantities
,
,
can be obtained in explicit form
Therefore, the dynamical system for the amplitude and position of the crest is
From Equation (
15) we have that the position of the crest of a solitary wave is described by the oscillator
Here, we formally consider the periodic external force to be as the one defined in Equation (
7). In fact, if the external force is broad in comparison with the soliton length, the asymptotic theory is valid for any function
—not only with vanishing ends. It works for periodic external forces with small values of
q. It can be shown that the equilibrium points of this dynamical system are
, where
k is an integer. Centers occur aligned with the crests of the external force while saddles are aligned with the troughs of the external force. Consequently, as we change the sign of
A, the centers become saddles and vice-versa. It is worth to mention that centers and saddles represent solitary waves that remain steady for all times, a closed orbit represents a trapped solitary wave, and a non-closed orbit corresponds to a solitary wave that propagates without reversing its direction. Therefore, a phase-portrait of the dynamical system (
15) qualitatively describes the behavior of the solitary wave and the external force interaction except for small corrections (order
).
Solutions of the dynamical system (
15) are represented by streamlines i.e., solutions are the level curves of the stream function
, which is given by
Figure 1 displays the typical phase portraits of the dynamical system (
15). We recall that a closed orbit illustrates a solitary wave that is trapped with no radiation due to its interaction with the external force while a non-closed orbit represents a solitary wave that propagates without reversing its direction. It is worth to mention that in
Figure 1 (left) each center is aligned with a crest of the external force while in
Figure 1 (right) each center is aligned with a trough of the external force. Although the asymptotic results presented here are limited to the weak-amplitude case, as it follows from Grimshaw and Pelinovsky [
17], qualitative results still hold for arbitrary amplitudes of the solitary waves and we do not reproduce them here.
3.2. Numerical Results
Equation (
8) is solved numerically in a periodic computational domain
with a uniform grid with
N points using a Fourier pseudospectral method with an integrating factor [
21]. The computational domain is taken large enough in order to prevent effects of the spatial periodicity. The time evolution is calculated through the Runge–Kutta fourth-order method with time step
. Typical computations are performed using
Fourier modes
and
in MATLAB. In order to verify the accuracy of the numerical solutions, simulations are compared using a different number of Fourier modes (
and
), and the results are the same. A study of the resolution of a similar numerical method can be found in Ref. [
7].
In order to compare the numerical solutions with the asymptotic predictions, we verify whether the equilibrium points of a dynamical system (
15) represent qualitative solutions of Equation (
8). In other words, we verify if a point near a saddle point represents a solitary wave that travels without reversing its direction and if a center point corresponds to a trapped solitary wave. Since there is a long list of parameters to be considered in the study of the interaction between a solitary wave and the external force (
7), we fix a few parameters, namely,
,
,
and
, where
n is an integer, which represents the number of waves in the interval
. Thus, large values of
n represent high frequencies while small values of
n represent low frequencies. Notice that with these choices of parameters, the initial solitary wave has amplitude
, as defined in Equation (
6). Additionally, the initial solitary waves are chosen to be with their crests located at
, where
. We recall that according to the dynamical system (
15), choosing
and the position of crest
, we have a saddle, and by choosing
and the position of crest
, we obtain a center.
Firstly, we consider the simplest case—the saddle points. To this end, we run a large number of simulations and observe that the solitary waves move past the external force without changing their directions. However, reflection is observed as a solitary wave pass over multiple bumps. The reflection decreases as the frequency of the external force increases, which causes a change in the amplitude of the solitary waves. This typical behavior is illustrated in
Figure 2 for different values of the parameter
n. Details of the variations in the amplitude of the solitary wave for the case
are given in
Figure 3. Notice that the amplitude of the solitary wave oscillates as it passes over each bump of the external force (see
Figure 3 (left)). Additionally, the solitary wave speed keeps oscillating in time due to the interaction with the external force as displayed in
Figure 3 (right). This dynamic is qualitatively represented by the non-closed orbits of the dynamical system (
15), see
Figure 1 (left). In fact, the fully numerical simulations and the dynamical system agree quantitatively well for small times. To see this, we recall that the asymptotic theory is obtained by truncating the terms of Equation (
9) at order
which is the same order of the amplitude variations depicted in
Figure 3 (left).
Now, we investigate if the center points of the dynamical system (
15) define trapped waves for Equation (
8). So, we let
q vary and compare the numerical results within the asymptotic framework. For large values of
n, a solitary wave barely feels the external force, consequently the solitary wave remains almost steady, resembling the equilibrium center point of the dynamical system (
15). As we decrease the values of
n, for instance
the solitary wave bounces back and forth close to its initial position for large times with little radiation being emanated. These results are in agreement with the ones predicted by the asymptotic theory and are illustrated in
Figure 4. We observe that large values of the parameter
n lead to small oscillations of the crest-position of the solitary wave, i.e., the larger
n is the closest the solitary wave remains to its initial position. In
Figure 5 (left) we display the amplitude vs. crest of the solitary wave position and its crest position along time is shown in
Figure 5 (right) for
. Notice that the amplitude dynamics in the amplitude vs. crest position phase resembles an unstable spiral. Meanwhile, we observe that the crest position oscillates by increasing in time. This indicates that the solitary wave might move away from its initial position. It worth to mention that it does not contradict the predictions of the asymptotic theory since both theories are only expected to agree well at small times. It is noteworthy that for small perturbations of
, the solitary waves still remain trapped close to their initial positions for large times. In particular, it shows that the asymptotic theory for a broad external force in the weak-amplitude solitary wave regime gives good results qualitatively.
When
n is small, numerical results differ from the asymptotic theory. In fact, the asymptotic method breaks for small values of
q. It occurs because for too small values of
q the forcing is proportional to
. Therefore, the solitary waves are not affected by the external force.
Figure 6 displays the evolution of two solitary waves for small values of
n. Notice that the solitary wave propagates to the left without reversing its direction. Moreover, the change in amplitude of these solitary waves is small (see
Figure 7 (left)) and the solitary wave speed is almost constant, as depicted in
Figure 7 (right). Initially, the amplitude of the solitary wave is adjusted to the external force and later it changes only slightly. Consequently, the solitary wave travels almost as a classical solitary wave solution of the unforced problem (
1).
It is worth to mention that similar results are observed for positive choices of the amplitude of the external force
. To illustrate this, we limit ourselves to show
Figure 8. As we compare the respective panels of
Figure 4 and
Figure 8 we see that they are all the same unless translations.