**1. Introduction**

Free surface waves of a two-dimensional channel flow for an inviscid and incompressible fluid have been studied when the rigid bottom of the channel has some obstacles [1,2]. Such free surface waves of shallow water with obstacles can be modeled by the forced Korteweg-de Vries (fKdV) equations [3,4]. There have been enormous applications of the Korteweg-de Vries (KdV) equation in various research areas such as mathematics, physics, fluid mechanics and hydrodynamics. The KdV equation with a forcing approximately describes the evolution of the free surface when a fluid flows over an obstacle. This forced KdV equation is related to the physical problems such as shallow-water waves over rocks, tsunami waves over obstacles, oceanic stratified flows encountering topographic obstacles, and acoustic waves on a crystal lattice. Moreover, it can be useful for other nonlinear differential equations including a nonlinear Schrodinger equation and a sine Gordon equation. These applications range from magnetohydrodynamic waves, geostrophic turbulence, atmosphere dynamics and the propagation of short laser pulses in optical fibers (see [5,6] and references therein).

The dynamics of the free surface wave motions is characterized by the Froude number *F* defined as *F* = *C*/ *<sup>g</sup>h*, where *C* is the upstream velocity and *g<sup>h</sup>* is the critical speed of a shallow water wave with the finite depth of the channel *h* and the gravity constant *g*. It can be expressed as *F* = 1 + *λ* with the perturbation measurement *λ* to the critical value 1 for a positive small parameter , then the forced Korteweg-de Vries (fKdV) equation can be obtained. Further, the flow can be classified

as supercritical if *λ* > 0 and subcritical if *λ* < 0. For supercritical flows, there is a critical value, *λ<sup>c</sup>*, such that solitary wave solutions exist when *λ* > *λc* and no such solutions exist when 0 < *λ* < *λ<sup>c</sup>*. Solitary wave solutions present some special features such as remaining stable without any changes in shape or speed when they evolve in time. Moreover, this critical value of *λc* is dependent on the bump configuration.

The bottom topography is a key determinant for the characteristic of wave motions. Interesting surface wave phenomena can be observed when a bottom topography is rather complicated. For one bump, there has been extensive analytic and numerical work [1,3,7,8]. Cusped stationary solitary wave solutions are founded analytically and their numerical stability is shown for the Dirac delta function as forcing [9]. It is well known that there exist two branches of stationary solitary wave solutions when a semi-circular bump is given [10]. One is the near zero solitary wave with lower amplitude and the other one is a higher amplitude one (sech2-shaped wave). Analytic stability of solitary wave solutions is discussed for the time-dependent forced KdV equation when a compactly supported symmetric bump is given [7]. It is proved that the near zero solution is stable with respect to time for a sufficiently large *λ* > 0. The other solitary wave solution (sech2-shaped solution) is unstable when it evolves in time.

Recently, more researchers pay attention to the cases where bottom configurations are more complicated [11,12]. For subcritical flows, the generalized hydraulic falls have been obtained when two bumps are given [13]. These solutions are characterized by a train of waves 'trapped' between two bumps satisfying the radiation conditions at the infinity. The steady surface flows over a step and a rectangular obstacle have been investigated for supercritical, subcritical waves and hydraulic falls [14]. Various stationary solutions are explored when the bottom configurations are complex including one positive bump, one negative hole, or combinations of both [15]. Stationary solutions of the forced KdV with obstacles with a negative hole have been illustrated as the amplitude and the width of a negative hole are varied [16].

Analytic and numeric stationary solutions of the forced KdV equations are studied when one bump or two bumps are given as forcing, which are in the form of sech2- or sech4-functions [17]. They also explored the stability of solitary waves and table-top solutions when they evolve in time. Multiple stationary supercritical solutions of the fKdV equation are discussed when two-semi-circular bumps are given [18]. Their results show that there exists only one supercritical positive solitary wave, which is stable when it evolves in time. Solitary wave solutions and their time evolutions are investigated when the rigid bottom has one negative hole [19,20]. Interestingly, there are five stationary solutions when a negative hole is given and only the near zero wave remains stable when they evolve in time [19].

In our previous study, stationary *trapped solitary wave solutions* have been studied for the forced Korteweg-de Vries equation when two bumps or two holes are given as the bottom configuration [21]. Some of multiple stationary solitary waves have been found and then their numerical stability has been investigated in the presence of two bumps [22]. Interestingly, multiple *trapped supercritical wave solutions* remain stable between two bumps for a very long time when they evolve in time. It is worth noting that the interplay between trapped solitary waves and two bumps plays a key role in determining their time evolutions. In this work, we extend the previous work by exploring the impact of different two-bump scenarios on the trapped solitary waves and their numerical stability. First, various stationary trapped solitary wave solutions of the forced KdV equation have been obtained in the presence of two bumps or two holes. As the bump size or the distance between two bumps are varied, stationary solitary waves are obtained. Next, we employ the semi-implicit finite difference method which has been developed for the homogeneous KdV equation [23]. This numerical method has been used to obtain the time evolution of the trapped stationary solitary waves. Our numerical results show that multiple *trapped solitary wave solutions* stay stable between two bumps for a very long time under various two-bump configurations. This indicates that the interplay between trapped

solitary waves and two bumps highlights the importance in determining the dynamics of trapped solitary waves.
