# Asymmetric Perfect Absorption and Lasing of Nonlinear Waves by a Complex δ-Potential

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## Abstract

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## 1. Introduction

## 2. The Model

## 3. The Main Results

#### 3.1. Symmetric and Asymmetric Perfectly Absorbed Flows

#### 3.2. Stability of Perfectly Absorbed Nonlinear Currents

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BEC | Bose–Einstein condensate |

CPA | coherent perfect absorption |

NLSE | nonlinear Scrödinger equation |

SS | spectral singularity |

## References

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**Figure 1.**Examples of orbits generated by the differential Equation (7) in the phase space $(\rho ,{\rho}_{x})$ for several different values of J. The heteroclinic orbits shown with red lines correspond to the dark soliton solution that is immune to the dissipation and bears zero background current $J=0$. Homoclinic orbits with $J=0.4,1,1.4$ correspond to perfectly absorbed flows with a density dip in a uniform background. Each homoclinic orbit passes through the saddle point $(\rho ,{\rho}_{x})=(R,0)$, where R is the background amplitude. Depending on the position of the point that corresponds to the dissipative spot, i.e., $x=0$ in our case, each orbit can represent either a symmetric or an asymmetric perfectly absorbed flow. In this figure, the background amplitude of nonlinear flows is fixed as $R=1$.

**Figure 2.**The existence diagram for solutions of three different types discussed in Section 3.1. The diagram is shown in the plane J vs. $\gamma $ for fixed background density $R=1$. The gray domain corresponds to asymmetric dips, whereas red and green lines correspond to the constant-amplitude solutions and symmetric dips, respectively.

**Figure 3.**(

**a**) Dependencies of the position of the amplitude dip ${x}_{0}$ defined by (11) on the absorption strength $\gamma $ for the fixed values of the background current: $J=1/2$ (red curve) and $J=1$ (blue curve). (

**b**) Squared amplitude of asymmetric states that exist at $\gamma =3/2$ with $J=1/2$ (red curve) and $J=1$ (blue curve). In both panels $R=1$. Only solutions with ${x}_{0}\ge 0$ are shown; there also coexist their mirror counterparts with ${x}_{0}\le 0$.

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**MDPI and ACS Style**

Zezyulin, D.A.; Konotop, V.V.
Asymmetric Perfect Absorption and Lasing of Nonlinear Waves by a Complex *δ*-Potential. *Symmetry* **2020**, *12*, 1675.
https://doi.org/10.3390/sym12101675

**AMA Style**

Zezyulin DA, Konotop VV.
Asymmetric Perfect Absorption and Lasing of Nonlinear Waves by a Complex *δ*-Potential. *Symmetry*. 2020; 12(10):1675.
https://doi.org/10.3390/sym12101675

**Chicago/Turabian Style**

Zezyulin, Dmitry A., and Vladimir V. Konotop.
2020. "Asymmetric Perfect Absorption and Lasing of Nonlinear Waves by a Complex *δ*-Potential" *Symmetry* 12, no. 10: 1675.
https://doi.org/10.3390/sym12101675