*3.1. Trapped Solitary Waves between Two Positive Bumps*

As well known, the time independent KdV equation without any forcing (i.e. *f*(*x*) = 0) has two exact solutions such as *η*1(*x*) ≡ 0 and *η*2(*x*) = <sup>2</sup>*λsech*<sup>2</sup>[(6*<sup>λ</sup>*)1/2(*x* − *<sup>x</sup>*0)/2] with a phase shift *x*0. Also, in the presence of a single positive symmetric forcing which satisfies a compact support condition, there are two symmetric solitary wave solutions: one is near *η*1(*x*) and the other one is near *η*2(*x*) for all *λ* > *λc* with a critical value *λc* > 0 [7]. Under two-bump configurations, with the similar argument, there exist at least two stationary solitary wave solutions, since the bump *h*(*x*; *a*, *P*1, *<sup>P</sup>*2) is positive and symmetric with a compact support. Recall that *h*(*x*; *a*, *P*1, *<sup>P</sup>*2) is defined as each bump is centered at −*a* and *a* and the bump height at each center is *P*1 = *P*2.

Figure 1 displays eight stationary wave solutions using the two-bump configuration *h*(*x*; 2, 1, 1) and *λ* = 1.5. The left panel shows two symmetric solitary wave solutions including the lower amplitude wave (blue dashed), which is the near zero solution denoted by *ηs*(*x*). Two non-symmetric wave solutions (with lower and higher amplitude, respectively) are shown in the middle panel. The right panel represents four stationary wave solutions which consist of two symmetric solutions and two non-symmetric ones. The blue dashed wave illustrated in the right panel of Figure 1 depicts the solitary wave solution which is trapped between two bumps similar to *η*2(*x*) (*sech*2-type wave is placed in the middle of two bumps). Let this be referred to as a *trapped solitary wave solution* and denoted by *ηT*(*x*). In this study, we focus on the dynamics of these trapped solitary waves, *ηT*(*x*) (a higher amplitude wave in the middle of two bumps) under various bump configurations.

**Figure 1.** Stationary wave solutions and a two-bump configuration are shown with *λ* = 1.5 and *h*(*x*; 2, 1, <sup>1</sup>). The left panel shows the stable solitary wave *ηs*(*x*) (the blue dashed wave) while the middle panel shows two non-symmetric waves. The right panel displays the trapped solitary wave solution *ηT*(*x*), the blue dashed wave between two bumps.

It is well known that the near zero wave solution *ηs*(*x*) is stable with respect to time when either single positive symmetric bump or two positive bumps is given. The solitary wave solutions with two higher amplitudes (the solid curves in the left panel of Figure 1) are moving out fast since they are placed right over the bumps and this makes them unstable in a very short time. However, the trapped solitary wave which is placed between two bumps (*ηT*(*x*) in the right panel of Figure 1) remains stable for a very long time as observed [22]. Here, the time evolution of the trapped solitary wave, *ηT*(*x*) is revisited in Figures 2 and 3. The unperturbed trapped solitary wave solution (*ηT*(*x*)) remains stable between two bumps until *t* = 250 in Figure 2. Figure 3 displays the time evolutions of perturbed trapped solitary wave solutions with +5% and −5% perturbation, respectively. Under ±5% perturbations, trapped solitary waves start moving out of two bumps around *t* = 70. It is worth noting that the trapped solitary wave *ηT*(*x*) consists of the unforced solitary wave *η*2(*x*) and the near zero solitary waves *ηs*(*x*) with two bumps. Hence, we can conjecture that the trapped solitary wave *ηT*(*x*) remains stable up to a certain time due to the stable interplay between *η*2(*x*) and *ηs*(*x*).

**Figure 2.** The time evolution of the trapped solitary wave solution is illustrated without any perturbation using *h*(*<sup>x</sup>*, 2; 1, 1) and *λ* = 1.5. The trapped solitary wave remains stable up to a long time up to *t* = 250.

x

**Figure 3.** The time evolution of the trapped solitary wave solution is shown using *h*(*<sup>x</sup>*, 2; 1, 1) and *λ* = 1.5 with +5% perturbation (**left**) and −5% perturbation (**right**), respectively. They start moving between two bumps and evolve out of two bumps around *t* = 70.

### *3.2. The Impact of the Bump Distance on the Stability of ηT*(*x*)

We investigate the impact of the distance between two bumps on the trapped solitary waves. Figure 4 illustrates three stationary trapped solitary wave solutions using three different bump distances between two bumps. All of their shapes are similar to *η*2(*x*) around *x* = 0 with the near zero wave just over each bump. The distance between two bumps are varied as 2, 4 and 6 using *h*(*<sup>x</sup>*, 2; 1, <sup>1</sup>), *h*(*<sup>x</sup>*, 3; 1, 1) and *h*(*<sup>x</sup>*, 4; 1, <sup>1</sup>), respectively. Their time evolutions are illustrated in Figures 5 and 6.

First, the time evolution of the unperturbed trapped solitary wave solution is displayed using *h*(*<sup>x</sup>*, 3; 1, 1) (the distance between two bumps is 4) in Figure 5. It is interesting to observe that the trapped solitary wave solution is stable up to a very long time (simulated until *t* = 500). This can be compared with the unperturbed result with *h*(*<sup>x</sup>*, 2; 1, 1) stable up to *t* = 250 in Figure 2. Next, the trapped solitary wave using *h*(*<sup>x</sup>*, 3; 1, 1) are perturbed with +5% and −5% in Figure 6. They move back and forth between two bumps, however, they still remain between two bumps for a very long time (the results shown until *t* = 100).

**Figure 4.** Stationary trapped solitary wave solutions and two-bump configurations are displayed using three bump distances *h*(*<sup>x</sup>*, 2; 1, <sup>1</sup>), *h*(*<sup>x</sup>*, 3; 1, 1) and *h*(*<sup>x</sup>*, 4; 1, <sup>1</sup>), respectively.

**Figure 5.** The time evolution of the trapped solitary wave solution is displayed with *h*(*<sup>x</sup>*, 3; 1, 1) and *λ* = 1.5. The unperturbed trapped solitary wave remains stable between two bumps for a very long time (simulated up to *t* = 500).

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**Figure 6.** The time evolution of the trapped solitary wave solution is shown for *h*(*<sup>x</sup>*, 3; 1, 1) and *λ* = 1.5 with +5% perturbation (**left**) and −5% perturbation (**right**), respectively. The perturbed trapped solitary waves bounce between two bumps for a long time (shown up to *t* = 100).

### *3.3. The Impact of the Bump Size on the Stability of ηT*(*x*)

The impact of the size of two bumps on the time evolutions for trapped solitary waves has been explored. Let us consider our baseline bump size to be 1 (the bump size is defined as the maximum height of the bump at the center, which is *P*1 = *P*2). Now, the bump size is varied from 0.5 to 2. As shown in Figure 7, each stationary trapped solitary wave has the shape of *η*2(*x*) around *x* = 0 and the near zero wave over each bump. Also, the distance between two bumps is varied as well; the left panel shows the result with the bump distance 2 and the right panel with the bump distance 4. The amplitude of near zero wave gets higher as the bump size becomes larger in both panels.

The time evolutions of the perturbed trapped solitary wave solution using *h*(*<sup>x</sup>*, 2; 0.5, 0.5) with +5% and −5% are displayed in Figure 8. Note that the perturbed trapped solitary wave solution stays much longer between two bumps when comparing with the results in Figure 4. That implies that the trapped solitary wave with the bump size 0.5 is more stable than the ones with the bump size 1. Now, using the larger bump size, Figure 9 illustrates that the perturbed trapped solitary wave solutions with the bump size 2 moves out of two bumps quickly than the results in Figure 4. This indicates that there is a critical bump size for trapped solitary waves to remain stable for a longer time. Therefore, we investigate the relationship between the bump size and the time until trapped solitary waves remain stable.

**Figure 7.** Stationary trapped solitary wave solutions are displayed using three different bump sizes, *P*1 = *P*2 = 0.5, *P*1 = *P*2 = 1 and *P*1 = *P*2 = 2 when the distance between two bumps is 2 (**left**) and 4 (**right**), respectively.

**Figure 8.** The time evolution of the trapped solitary wave solution using *h*(*<sup>x</sup>*, 2; 0.5, 0.5) and *λ* = 1.5 with +5% perturbation (**left**) and −5% perturbation (**right**), respectively. They start moving between two bumps and evolve out of two bumps around *t* = 340 and *t* = 300, respectively.

**Figure 9.** The time evolution of the trapped solitary wave solution using *h*(*<sup>x</sup>*, 2; 2, 2) and *λ* = 1.5 with +5% perturbation (**left**) and −5% perturbation (**right**), respectively. They start moving between two bumps and evolve out of two bumps around *t* = 50 and *t* = 40, respectively.

The left panel of Figure 10 displays the time when the trapped solitary wave solutions move out of two bumps as the bump size is varied. The time is measured when trapped solitary waves under +5% (solid) or −5% (dotted) perturbation start moving out of two bumps. The time decreases when the bump size increases from 0.5 to 2 so that it takes the minimum around the bump size 2, then the time increases again significantly as the bump size increases. Next, we investigate the relationship between the bump distance and the time until trapped solitary waves remain stable. The right panel of Figure 10 illustrates the time when the trapped solitary wave solutions move out of two bumps as the bump distance is varied. Again, the time is measured when trapped solitary waves under +5% (solid) or −5% (dotted) perturbation start moving out of two bumps. As seen in the right panel, the time increases as the distance between two bumps increases linearly.

**Figure 10.** The impact of the bottom configurations is shown on the time when perturbed trapped solitary waves start evolving out of between two bumps (the left panel for the bump size and the right panel for the bump distance).
