**4. Conclusions**

In this work, the focus has been on the trapped solitary wave solutions of the forced KdV and their numerical stability in the presence of two bumps. We use the Newton method incorporating artificial boundary conditions to find various trapped solitary waves in the presence of two positive bumps and two negative holes. This method provides a good alternative to find various stationary wave solutions under arbitrary two-bump configurations when they are close enough (the distance between two bumps are not large) [21]. The semi-implicit finite difference method has been employed to investigate the numerical stability of these trapped solitary waves. Our previous results indicate that the number of stationary wave solutions increases as the bump distance increases and there exist multiple trapped solitary waves under different two-bump configurations [22]. In this study, we have extended the results under more various two bump configurations. Specifically, the impact of the bump distance and the bump size on numerical stability of trapped solitary waves have been investigated.

For both two positive bumps and two negative holes, the near zero solution is stable when it evolves in time. Moreover, it is worth to observe that a *trapped wave solution* remains stable for a very longer time between two positive bumps. As the distance between two bumps increases, trapped solitary waves stay longer in a straightforward fashion. On the other hand, the relationship for the bump size and the time trapped solitary waves to remain between the bumps is not trivial. There is a critical bump size for the trapped solitary to be stable for a longer time. Furthermore, for two negative holes, the trapped solitary waves do not remain for a long time as two positive bumps are given. Their time evolutions are characterized by the interplay between trapped solitary waves and two-bump configurations.

However, for non-symmetric bumps (two bumps or two holes with different sizes), trapped solitary waves become non-symmetric as well and they move out of two bumps in a much shorter time. Non-symmetry becomes even more significant in stationary solitary waves for the combined bottom configuration of a bump and a hole. They lose stability too fast to measure the length of time where solitary waves are stable. This highlights the importance of symmetry in the bumps and solitary waves for their stability. We find that the interplay between trapped solitary waves and two bumps becomes stronger so that they remain stable up to a certain time when two positive bumps are given. Hence, we can conjecture that two positive bumps provide a trap for some trapped wave solutions to be stable for a certain finite time.

**Acknowledgments:** This work was supported by the National Research Foundation of Korea (NRF) gran<sup>t</sup> funded by the Korean governmen<sup>t</sup> (MSIP) (NRF-2015R1C1A2A01054944).

**Conflicts of Interest:** The authors declare no conflict of interest.
