# Dynamics of Trapped Solitary Waves for the Forced KdV Equation

## Abstract

**:**

## 1. Introduction

## 2. The Forced KdV Equation

#### 2.1. Model Equation with Two-Bump Configurations

#### 2.2. Semi-Implicit Finite Difference Method

## 3. Numerical Simulations

#### 3.1. Trapped Solitary Waves between Two Positive Bumps

#### 3.2. The Impact of the Bump Distance on the Stability of ${\eta}_{T}\left(x\right)$

#### 3.3. The Impact of the Bump Size on the Stability of ${\eta}_{T}\left(x\right)$

#### 3.4. Trapped Solitary Waves between Two Negative Holes

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | linear dichroism |

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**Figure 1.**Stationary wave solutions and a two-bump configuration are shown with $\lambda =1.5$ and $h(x;2,1,1)$. The left panel shows the stable solitary wave ${\eta}_{s}\left(x\right)$ (the blue dashed wave) while the middle panel shows two non-symmetric waves. The right panel displays the trapped solitary wave solution ${\eta}_{T}\left(x\right)$, the blue dashed wave between two bumps.

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

**Figure 3.**The time evolution of the trapped solitary wave solution is shown using $h(x,2;1,1)$ and $\lambda =1.5$ with $+5\%$ perturbation (

**left**) and $-5\%$ perturbation (

**right**), respectively. They start moving between two bumps and evolve out of two bumps around $t=70$.

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

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

**Figure 6.**The time evolution of the trapped solitary wave solution is shown for $h(x,3;1,1)$ and $\lambda =1.5$ 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 $t=100$).

**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 $h(x,2;0.5,0.5)$ and $\lambda =1.5$ 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 $h(x,2;2,2)$ and $\lambda =1.5$ 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.

**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).

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

**Figure 13.**The time evolution of trapped solitary wave solution is displayed with $\lambda =1.5$ and $h(x,2;-1,-1)$ (

**left**) and $h(x,3;-1,-1)$ (

**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 $\lambda =1.5$ and $h(x,3;-0.5,-0.5)$ (

**left**) and $h(x,3;-2,-2)$ (

**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).

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Lee, S.
Dynamics of Trapped Solitary Waves for the Forced KdV Equation. *Symmetry* **2018**, *10*, 129.
https://doi.org/10.3390/sym10050129

**AMA Style**

Lee S.
Dynamics of Trapped Solitary Waves for the Forced KdV Equation. *Symmetry*. 2018; 10(5):129.
https://doi.org/10.3390/sym10050129

**Chicago/Turabian Style**

Lee, Sunmi.
2018. "Dynamics of Trapped Solitary Waves for the Forced KdV Equation" *Symmetry* 10, no. 5: 129.
https://doi.org/10.3390/sym10050129