*3.4. Trapped Solitary Waves between Two Negative Holes*

In this subsection, we present the dynamics of trapped solitary waves when two negative holes are given as the bottom configuration. As studied in [19], there are five stationary wave solutions including the near zero wave (a negative solitary wave) and the near *η*2(*x*) wave in the presence of one negative bump or hole. Unlike the solitary wave of a positive bump, the near zero solitary wave exists for all *λ* > 0 when a negative bump is given. Moreover, the near zero wave is stable and the near *η*2(*x*) wave is unstable when they evolve in time [19]. In the presence of two symmetric negative

holes, the number of stationary solitary wave solutions increase (the larger the distance between two holes, the more various trapped solitary waves [21]).

First, stationary trapped solitary wave solutions between two holes are obtained as the distance between two holes is varied. In Figure 11, three stationary trapped solitary wave solutions are displayed under three different bump distance using *h*(*<sup>x</sup>*, 2; −1, <sup>−</sup><sup>1</sup>), *h*(*<sup>x</sup>*, 3; −1, −<sup>1</sup>) and *h*(*<sup>x</sup>*, 4; −1, <sup>−</sup><sup>1</sup>), respectively. Each trapped solitary wave has the shape of *η*2(*x*) around *x* = 0 and the near zero negative wave over each hole. Next, Figure 12 illustrates stationary trapped solitary waves as the hole depth is varied, −0.5, −1 and −2. Further, the left panel of Figure 12 shows the trapped solitary waves using the hole distance 2 and the right panel shows the trapped solitary waves using the hole distance 4.

**Figure 11.** Stationary trapped solitary wave solutions are shown using three two-hole configurations with three different distances, *h*(*<sup>x</sup>*, 2; −1, <sup>−</sup><sup>1</sup>), *h*(*<sup>x</sup>*, 3; −1, −<sup>1</sup>) and *h*(*<sup>x</sup>*, 4; −1, <sup>−</sup><sup>1</sup>), respectively.

**Figure 12.** Stationary trapped solitary wave solutions are displayed using three different hole sizes, *P*1 = *P*2 = −0.5, *P*1 = *P*2 = −1 and *P*1 = *P*2 = −2 when the distance between two holes is 2 (**left**) and 4 (**right**), respectively.

Figures 13 and 14 illustrate the time evolutions of unperturbed trapped solitary wave solutions. As seen in both figures, the trapped solitary waves move out of two holes much faster than the cases of two symmetric positive bumps. In Figure 13, the trapped solitary wave stays between two holes up to *t* = 100 with *h*(*<sup>x</sup>*, 3; −1, −<sup>1</sup>) (right) while it stays only up to *t* = 20 with *h*(*<sup>x</sup>*, 2; −1, −<sup>1</sup>) (left). This is consistent with the results for two positive bumps; the larger the distance between two holes, the longer the trapped solitary wave solution remains stable between two holes. Similarly, as we vary the depth of two holes, the time evolutions of trapped solitary waves are changing as well in a complex way. As the depth of two holes is reduced half *h*(*<sup>x</sup>*, 3; −0.5, <sup>−</sup>0.5), the trapped solitary wave remains between two holes for a longer time (see the left panel of Figure 14). The depth of two holes is

increased double *h*(*<sup>x</sup>*, 3; −2, <sup>−</sup><sup>2</sup>), the trapped solitary wave remains between two holes for a shorter time (see the right panel of Figure 14). Again, this is consistent with the results for two positive bumps.

**Figure 13.** The time evolution of trapped solitary wave solution is displayed with *λ* = 1.5 and *h*(*<sup>x</sup>*, 2; −1, −<sup>1</sup>) (**left**) and *h*(*<sup>x</sup>*, 3; −1, −<sup>1</sup>) (**right**), respectively. The unperturbed trapped solitary wave remains stable between two holes for a very short time *t* = 20 (see the **left** panel). It remains for a longer time when the distance increases (stable up to *t* = 100 in the **right** panel).

**Figure 14.** The time evolution of trapped solitary wave solution with *λ* = 1.5 and *h*(*<sup>x</sup>*, 3; −0.5, −0.5) (**left**) and *h*(*<sup>x</sup>*, 3; −2, −<sup>2</sup>) (**right**), respectively. The unperturbed trapped solitary wave remains stable between two holes up to *t* = 120 (see the **left** panel). It remains for a shorter time when the depth of holes increases to −2 (stable up to *t* = 60 in the **right** panel).
