# Soliton–Breather Interaction: The Modified Korteweg–de Vries Equation Framework

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Breather and Soliton Solutions of the mKdV Equation

_{0}and f

_{0}can be removed by the appropriate coordinate and time conversions. The physical meaning of the other constants is as follows: p affects the number of waves in a packet, and q determines the breather’s amplitude. If the q >> p, a breather has few cycles and resembles the superposition of the two mKdV solitons (2). A breather containing many oscillations can be interpreted as an envelope soliton of the nonlinear Schrödinger equation.

## 3. Breather’s Extrema

^{−5}and the calculations stopped when the distance between waves after their interactions became the same as at t = 0. During the calculations, the maximum amplitude of the resulting pulse, and higher moments of the wave field (${M}_{3,4}\left(t\right)={{\displaystyle \int}}_{-\infty}^{+\infty}{u}^{3,4}\left(x,t\right)dx$) were registered.

^{−14}and 10

^{−7}, respectively.

#### 3.1. Interaction Process

#### 3.2. Extrema of the Wave Fields

_{0}= −41.1 and x

_{0}= −99 correspondingly), see Figure 2. Since the two regimes differ significantly, the horizontal scale is different in the two panels on the Figure. The pulse’s shapes at the “optimal” interacting moment are shown in Figure 3. These graphs are symmetrical unlike the other experiments.

#### 3.3. Moments of the Wave Fields

_{3}= M

_{3_sol}in the initial moment and it is equal to 1.57. The fourth moment consists of both, soliton and breather components: M

_{4}= M

_{4_sol}+ M

_{4_br}= 2.3 (Figure 4a). Since, the breather is not horizontally symmetric in the initial moment in the second series of experiments, the analytical expressions satisfy the time moment t = 30.75, when the soliton and the breather are still far from each other, and the breather is represented by the sum of two identical solitons with different polarity. The moments are M

_{3}= 1.57, M

_{4}= 1.73 (Figure 4b).

_{n}* = (M

_{n_max}− M

_{n_min})/M

_{n_max}, n = 3, 4) for the first set of the experiments showed that the maximum variations correspond to the optimal focusing case (Figure 5a). For some wave phases, the tails of M

_{3}* are even higher than those of M

_{4}*. The behavior of these variations is non-monotonic in the second series of experiments (Figure 5b). Deviations of M

_{3}* are always larger than M

_{4}*. When the maximum elevation of resultant pulse is the smallest in the series, there is huge depression of M

_{4}*. Its biggest variation corresponds to the “optimal interaction” case.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Soliton–breather interaction with breather parameters: (

**a**) p = 2, q = 0.25, (

**b**) p = 0.05, q = 0.25.

**Figure 2.**Maximum maximorum of the wave fields on the soliton phase shift x

_{0}: (

**a**) in the first series of experiments (p = 2, q = 0.25), (

**b**) in the second series of experiments (p = 0.05, q = 0.25).

**Figure 3.**The shapes of the resulting pulse during the “optimal” soliton–breather interaction: (

**a**) in the first series of experiments (p = 2, q = 0.25), (

**b**) in the second series of experiments (p = 0.05, q = 0.25).

**Figure 4.**3rd and 4th wave field moments during one experiment with the maximum wave field amplification from: (

**a**) the first series of experiments (p = 2, q = 0.25), (

**b**) from the second series of experiments (p = 0.05, q = 0.25).

**Figure 5.**Relative variations of the 3rd and 4th wave field moments on the soliton phase shift: (

**a**) in the first series of experiments (p = 2, q = 0.25), (

**b**) in the second series of experiments (p = 0.05, q = 0.25).

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Didenkulova, E.; Pelinovsky, E.
Soliton–Breather Interaction: The Modified Korteweg–de Vries Equation Framework. *Symmetry* **2020**, *12*, 1445.
https://doi.org/10.3390/sym12091445

**AMA Style**

Didenkulova E, Pelinovsky E.
Soliton–Breather Interaction: The Modified Korteweg–de Vries Equation Framework. *Symmetry*. 2020; 12(9):1445.
https://doi.org/10.3390/sym12091445

**Chicago/Turabian Style**

Didenkulova, Ekaterina, and Efim Pelinovsky.
2020. "Soliton–Breather Interaction: The Modified Korteweg–de Vries Equation Framework" *Symmetry* 12, no. 9: 1445.
https://doi.org/10.3390/sym12091445