# Dynamics of Benjamin–Ono Solitons in a Two-Layer Ocean with a Shear Flow

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## Abstract

**:**

## 1. Introduction

## 2. Dispersion Relations for a Two-Layer Fluid with a Shear Flow

#### Analysis of the Dispersion Relation

## 3. Generalized Benjamin–Ono Equations with Dissipative Terms

#### 3.1. Linear Evolution Equations

#### 3.2. Derivation of the Nonlinear Term in the BO Equation

## 4. Dynamics of a Solitary Wave Under the Influence of Dissipation

#### 4.1. Soliton Decay Due to Reynolds-Type Dissipation

#### 4.2. Soliton Dynamics under the Influence of Viscosity in a Moving Upper Layer

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Dynamics of Periodic Waves within the genBO Equation with Dissipative Terms

**Figure A1.**(Color online). Shapes of periodic solutions of the BO equation of the same wavelength ($\Lambda -2\pi $) and zero mean values but with different parameter $\gamma $. Line 1—quasi-sinusoidal wave with $\gamma =0.1$ (wave amplitude multiplied by 10 to make it clearly visible); line 2—a periodic sequence of quasi-soliton waves with $\gamma =0.99$.

**Figure A2.**(Color online). Dependence of parameter $\gamma $ on normalized time ($\tau ={\delta}_{2}t{(\pi /\Lambda )}^{3}$). Line 1 shows the increase in parameter $\gamma $ from ${\gamma}_{0}=0.01$ to ${\gamma}_{lim}=1$ for the supercritical case of a flow. Line 2 shows the decrease in parameter $\gamma $ from ${\gamma}_{0}=0.95$ to zero for the subcritical case of a flow.

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**Figure 1.**Sketch of a fluid flow in the two-layer model with infinitely deep and immovable lower layer.

**Figure 2.**(Color online). Real parts of approximate dispersion relations (25) in terms of dimensionless variables ${\Omega}_{1,2}\left(\kappa \right)={\omega}_{1,2}^{0}{h}_{1}/{c}_{1}$, where $\kappa =k{h}_{1}$ and $a=0.999$. Lines 1 and 2 pertain to ${\omega}_{1}^{0}$ and ${\omega}_{2}^{0}$, respectively, with $\mathrm{Fr}=0$. Lines 3 and 4 pertain to ${\omega}_{1}^{0}$ and ${\omega}_{2}^{0}$, respectively, with $\mathrm{Fr}=1.1$.

**Figure 3.**(Color online). Soliton amplitude’s dependence on normalized time ($t/\tau $) for Reynolds dissipation. Solid line—theoretical dependence (62); dots—numerical data; dashed line—asymptotic dependence $\eta \sim {(t/\tau )}^{-1/2}$.

**Figure 4.**(Color online). Solitary wave profile (line 1) at $t/\tau $ = 3000 for BO Equation (54) with Reynolds-type dissipation. Line 2 represents a BO soliton of the same amplitude as the leading pulse shown by line 1.

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**MDPI and ACS Style**

Negi, P.; Sahoo, T.; Singh, N.; Stepanyants, Y.
Dynamics of Benjamin–Ono Solitons in a Two-Layer Ocean with a Shear Flow. *Mathematics* **2023**, *11*, 3399.
https://doi.org/10.3390/math11153399

**AMA Style**

Negi P, Sahoo T, Singh N, Stepanyants Y.
Dynamics of Benjamin–Ono Solitons in a Two-Layer Ocean with a Shear Flow. *Mathematics*. 2023; 11(15):3399.
https://doi.org/10.3390/math11153399

**Chicago/Turabian Style**

Negi, Pawan, Trilochan Sahoo, Niharika Singh, and Yury Stepanyants.
2023. "Dynamics of Benjamin–Ono Solitons in a Two-Layer Ocean with a Shear Flow" *Mathematics* 11, no. 15: 3399.
https://doi.org/10.3390/math11153399