# Symmetry Breakings in Dual-Core Systems with Double-Spot Localization of Nonlinearity

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Solutions Preserving the Spatial Symmetry in the Cores

#### 3.1. The Basic Model

#### 3.2. Review of Related Models

## 4. The Case of the Symmetry Maintained between the Cores

## 5. Double Symmetry Breaking in Two Dimensions

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Huang, W.P. Coupled-mode theory for optical waveguides: An overview. J. Opt. Soc. Am. A
**1994**, 11, 963–983. [Google Scholar] [CrossRef] - Digonnet, M.J.F.; Shaw, H.J. Analysis of a tunable single-mode optical fiber coupler. IEEE J. Quant. Electr.
**1982**, 18, 746–754. [Google Scholar] [CrossRef] - Trillo, S.; Wabnitz, S.; Wright, E.M.; Stegeman, G.I. Soliton switching in fiber nonlinear directional couplers. Opt. Lett.
**1988**, 13, 672–674. [Google Scholar] [CrossRef] [PubMed] - Harsoyono, H.; Siregar, R.E.; Tjia, M.O. A study of nonlinear coupling between two identical planar waveguides. J. Nonlinear Opt. Phys. Mater.
**2001**, 10, 233–247. [Google Scholar] [CrossRef] - Strecker, K.E.; Partridge, G.B.; Truscott, A.G.; Hulet, R.G. Bright matter wave solitons in Bose-Einstein condensates. New J. Phys.
**2003**, 5, 73.1. [Google Scholar] [CrossRef] - Gubeskys, A.; Malomed, B.A. Symmetric and asymmetric solitons in linearly coupled Bose-Einstein condensates trapped in optical lattices. Phys. Rev. A
**2007**, 75, 063602. [Google Scholar] [CrossRef] - Salasnich, L.; Malomed, B.A.; Toigo, F. Competition between the symmetry breaking and onset of collapse in weakly coupled atomic condensates. Phys. Rev. A
**2010**, 81, 045603. [Google Scholar] [CrossRef] - Matuszewski, M.; Malomed, B.A.; Trippenbach, M. Spontaneous symmetry breaking of solitons trapped in a double-channel potential. Phys. Rev. A
**2007**, 75, 063621. [Google Scholar] [CrossRef] - Pitaevskii, L.; Stringari, S. Bose-Einstein Condensation; Clarendon: Oxford, UK, 2003. [Google Scholar]
- Jensen, S.M. The nonlinear coherent coupler. IEEE J. Quant. Electr.
**1982**, 18, 1580–1583. [Google Scholar] [CrossRef] - Maier, A.A. Optical transistors and bistable devices utilizing nonlinear transmission of light in systems with unidirectional coupled waves. Sov. J. Quant. Electron.
**1982**, 12, 1490–1494. [Google Scholar] [CrossRef] - Friberg, S.R.; Silberberg, Y.; Oliver, M.K.; Andrejco, M.J.; Saifi, M.A.; Smith, P.W. Ultrafast all-optical switching in dual-core fiber nonlinear coupler. Appl. Phys. Lett.
**1987**, 51, 1135–1137. [Google Scholar] [CrossRef] - Friberg, S.R.; Weiner, A.M.; Silberberg, Y.; Sfez, B.G.; Smith, P.S. Femtosecond switching in dual-core-fiber nonlinear coupler. Opt. Lett.
**1988**, 13, 904–906. [Google Scholar] [CrossRef] [PubMed] - Heatley, D.R.; Wright, E.M.; Stegeman, G.I. Soliton coupler. Appl. Phys. Lett.
**1988**, 53, 172–174. [Google Scholar] [CrossRef] - Królikowski, W.; Kivshar, Y.S. Soliton-based optical switching in waveguide arrays. J. Opt. Soc. Am. B
**1996**, 13, 876–887. [Google Scholar] [CrossRef] - Tsang, S.C.; Chiang, K.S.; Chow, K.W. Soliton interaction in a two-core optical fiber. Opt. Commun.
**2004**, 229, 431–439. [Google Scholar] [CrossRef] - Uzunov, I.M.; Muschall, R.; Gölles, M.; Kivshar, Y.S.; Malomed, A.; Lederer, F. Pulse switching in nonlinear fiber directional couplers. Phys. Rev. E
**1995**, 51, 2527–2537. [Google Scholar] [CrossRef] - Lederer, F.; Stegeman, G.I.; Christodoulides, D.N.; Assanto, G.; Segev, M.; Silberberg, Y. Discrete solitons in optics. Phys. Rep.
**2008**, 463, 1–126. [Google Scholar] [CrossRef] - Snyder, A.W.; Mitchell, D.J.; Poladian, L.; Rowland, D.R.; Chen, Y. Physics of nonlinear fiber couplers. J. Opt. Soc. Am. B
**1991**, 8, 2102–2112. [Google Scholar] [CrossRef] - Wright, E.M.; Stegeman, G.I.; Wabnitz, S. Solitary-wave decay and symmetry-breaking instabilities in two-mode fibers. Phys. Rev. A
**1989**, 40, 4455. [Google Scholar] [CrossRef] - Paré, C.; Florjańczyk, M. Approximate model of soliton dynamics in all-optical fibers. Phys. Rev. A
**1990**, 41, 6287–6295. [Google Scholar] [CrossRef] [PubMed] - Maimistov, A.I. Propagation of a light pulse in nonlinear tunnel-coupled optical waveguides. Kvantovaya Elektron. (Moscow)
**1991**, 18, 758–761. [Google Scholar] [CrossRef] - Chu, P.L.; Malomed, B.A.; Peng, G.D. Soliton switching and propagation in nonlinear fiber couplers: Analytical results. J. Opt. Soc. Am. B
**1993**, 10, 1379–1385. [Google Scholar] [CrossRef] - Akhmediev, N.; Ankiewicz, A. Novel soliton states and bifurcation phenomena in nonlinear fiber couplers. Phys. Rev. Lett.
**1993**, 70, 2395–2398. [Google Scholar] [CrossRef] [PubMed] - Soto-Crespo, J.M.; Akhmediev, N. Stability of the soliton states in a nonlinear fiber coupler. Phys. Rev. E
**1993**, 48, 4710–4715. [Google Scholar] [CrossRef] [Green Version] - Malomed, B.A.; Skinner, I.M.; Chu, P.L.; Peng, G.D. Symmetric and asymmetric solitons in twin-core nonlinear optical fibers. Phys. Rev. E
**1996**, 53, 4084–4091. [Google Scholar] [CrossRef] - Trillo, S.; Stegeman, G.; Wright, E.; Wabnitz, S. Parametric amplification and modulational instabilities in dispersive nonlinear directional couplers with relaxing nonlinearity. J. Opt. Soc. Am. B
**1989**, 6, 889–900. [Google Scholar] [CrossRef] - Tasgal, R.S.; Malomed, B.A. Modulational instabilities in the dual-core nonlinear optical fiber. Phys. Scr.
**1999**, 60, 418–422. [Google Scholar] [CrossRef] - Chiang, K.S. Intermodal dispersion in two-core optical fibers. Opt. Lett.
**1995**, 20, 997–999. [Google Scholar] [CrossRef] [PubMed] - Romagnoli, M.; Trillo, S.; Wabnitz, S. Soliton switching in nonlinear couplers. Opt. Quant. Electron.
**1992**, 24, S1237–S1267. [Google Scholar] [CrossRef] - Malomed, B.A. Variational methods in fiber optics and related fields. Progr. Opt.
**2002**, 43, 71–193. [Google Scholar] - Malomed, B.A. Solitons and nonlinear dynamics in dual-core optical fibers. In Handbook of Optical Fibers; Peng, G.-D., Ed.; Springer: Berlin, Germany, 2018. [Google Scholar]
- Iooss, G.; Joseph, D.D. Elementary Stability and Bifurcation Theory; Springer: Berlin, Germany, 1990. [Google Scholar]
- Hukriede, J.; Runde, D.; Kip, D. Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides. J. Phys. D
**2003**, 36, R1. [Google Scholar] [CrossRef] - Clark, L.W.; Ha, L.; Xu, C.; Chin, C. Quantum dynamics with spatiotemporal control of interactions in a stable Bose-Einstein condensate. Phys. Rev. Lett.
**2015**, 115, 155301. [Google Scholar] [CrossRef] [PubMed] - Malomed, B.A. (Ed.) Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations; Springer: Berlin/Heildelberg, Germany, 2013; ISBN 978-3-642-21206-2. [Google Scholar]
- Li, Y.; Malomed, B.A.; Feng, M.; Zhou, J. Double symmetry breaking of solitons in one-dimensional virtual photonic crystals. Phys. Rev. A
**2011**, 83, 053832. [Google Scholar] [CrossRef] - Mayteevarunyoo, T.; Malomed, B.A.; Dong, G. Spontaneous symmetry breaking in a nonlinear double-well structure. Phys. Rev. A
**2008**, 78, 053601. [Google Scholar] [CrossRef] - Salasnich, L.; Parola, A.; Reatto, L. Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates. Phys. Rev. A
**2002**, 65, 043614. [Google Scholar] [CrossRef] - Mateo, A.M.; Delgado, V. Effective mean-field equations for cigar-shaped and disk-shaped Bose-Einstein condensates. Phys. Rev. A
**2008**, 77, 013617. [Google Scholar] [CrossRef] - Sakaguchi, H.; Malomed, B.A. Symmetry breaking of solitons in two-component Gross-Pitaevskii equations. Phys. Rev. E
**2011**, 83, 036608. [Google Scholar] [CrossRef] [PubMed] - Acus, A.; Malomed, B.A.; Shnir, Y. Spontaneous symmetry breaking of binary fields in a nonlinear double-well structure. Physica D
**2012**, 241, 9872. [Google Scholar] [CrossRef] - Zhou, X.; Zhang, S.; Zhou, Z.; Malomed, B.A.; Pu, H. Bose-Einstein condensation on a ring-shaped trap with nonlinear double-well potential. Phys. Rev. A
**2012**, 85, 023603. [Google Scholar] [CrossRef] - Hung, N.V.; Zin, P.; Trippenbach, M.; Malomed, B.A. Two-dimensional solitons in media with stripe-shaped nonlinearity Modulation. Phys. Rev. E
**2010**, 82, 046602. [Google Scholar] [CrossRef] [PubMed] - Milburn, G.J.; Corney, J.; Wright, E.M.; Walls, D.F. Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential. Phys. Rev. A
**1997**, 55, 4318–4324. [Google Scholar] [CrossRef] [Green Version] - Smerzi, A.; Fantoni, S.; Giovanazzi, S.; Shenoy, S.R. Quantum coherent atomic tunneling between two trapped Bose-Einstein condensates. Phys. Rev. Lett.
**1997**, 79, 4950–4953. [Google Scholar] [CrossRef] - Chiofalo, M.L.; Succi, S.; Tosi, M.P. Ground state of trapped interacting Bose-Einstein condensates by an explicit imaginary-time algorithm. Phys. Rev. E
**2000**, 62, 7438–7444. [Google Scholar] [CrossRef] - Malomed, B.A.; Kevrekidis, P.G. Discrete vortex solitons. Phys. Rev. E
**2001**, 64, 026601. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Possible solutions of (11) plotted as the wavefunctions $\psi $ (red, solid) as well as $\varphi $ (blue, dashed) corresponding to the cases when: (

**a**) both symmetries are conserved; (

**b**) the symmetry between both cores is broken; (

**c**) only intra-core-symmetry is broken and (

**d**) both symmetries are broken. These wavefunctions are numerical solutions of (11) for $\mu =-0.9$ and $\kappa =0.5$.

**Figure 2.**The phase diagram, in the $\left(\kappa ,\left|\mu +\kappa \right|\right)$ plane, illustrating regions where solutions of different types exist. Solutions with unbroken symmetry exist everywhere. Above horizontal line ${\mu}_{+}=0.06$, in the horizontally shaded region, one additionally finds solutions with broken intra-channel symmetry. In the vertically shaded region, one observes solutions with broken symmetry between the channels (u and v). In the pink region, there exists solutions with both symmetries simultaneously broken. Note that there is a small triangular region where only fully symmetric solutions and ones with both symmetries broken can be found (the unshaded pink area).

**Figure 3.**Stationary states for a fixed value of the inter-core coupling constant, $\kappa =0.05$. Panels (

**a**) and (

**b**): the energy and norm of the fully symmetric states (black), ones with broken inter-core symmetry (orange), and states with broken $x\leftrightarrow -x$ symmetry in each channel (blue), as functions of the chemical potential. The inset in (

**a**) displays details of the emergence of the asymmetric states from the symmetric one. Panel (

**c**) displays the corresponding symmetry-breaking bifurcations by means of asymmetry parameters ${\mathsf{\Theta}}_{1}$ and ${\mathsf{\Theta}}_{2}$ (the orange and blue curves, respectively, see Equations (12) and (13), plotted versus the total norm.

**Figure 4.**Stationary states for fixed value of $\kappa =0.2$. Panels (

**a**) and (

**b**): energy and norm of the symmetric state (black), states with broken inter-core symmetry (orange), and states with broken $x\leftrightarrow -x$ symmetry in each channel (blue), vs. the chemical potential. The inset in (

**a**) displays details of the emergence of the asymmetric states from the symmetric one. In panel (

**c**), the corresponding symmetry-breaking bifurcations are shown by means of asymmetry parameters ${\mathsf{\Theta}}_{1}$ and ${\mathsf{\Theta}}_{2}$ (the orange and blue curves, respectively, see Equations (12) and (13), plotted versus the total norm.

**Figure 5.**Norm (

**a**) and energy (

**b**) of the symmetric, antisymmetric and asymmetric states as a function of the chemical potential in the model with the nonlinearity modulation in the form of the single delta-function, as defined by Equation (18). Solid and dashed lines denote stable and unstable states, respectively.

**Figure 6.**Asymmetry parameter $\mathsf{\Theta}$, defined as per Equation (24), as a function of the chemical potential (

**a**) and total norm (

**b**). Solid and dashed lines correspond to stable and unstable states, respectively.

**Figure 7.**The combination of the linear-potential double-trough trapping acting along the x direction, and the localized nonlinearity pattern shown by the striped structure. We address double symmetry breaking of two-dimensional solitons in this setting.

**Figure 8.**Contour plots of the local density illustrating stable two-dimensional self-trapped states. Panels (

**a**) and (

**b**): fully symmetric ones. (

**c**) and (

**d**): States with broken symmetry between the two cores, and between the two nonlinear spots, respectively. States featuring the breaking of both symmetries are displayed in panels (

**e**) and (

**f**).

© 2018 by the authors. 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/).

## Share and Cite

**MDPI and ACS Style**

Zegadlo, K.B.; Hung, N.V.; Ramaniuk, A.; Trippenbach, M.; Malomed, B.A.
Symmetry Breakings in Dual-Core Systems with Double-Spot Localization of Nonlinearity. *Symmetry* **2018**, *10*, 156.
https://doi.org/10.3390/sym10050156

**AMA Style**

Zegadlo KB, Hung NV, Ramaniuk A, Trippenbach M, Malomed BA.
Symmetry Breakings in Dual-Core Systems with Double-Spot Localization of Nonlinearity. *Symmetry*. 2018; 10(5):156.
https://doi.org/10.3390/sym10050156

**Chicago/Turabian Style**

Zegadlo, Krzysztof B., Nguyen Viet Hung, Aleksandr Ramaniuk, Marek Trippenbach, and Boris A. Malomed.
2018. "Symmetry Breakings in Dual-Core Systems with Double-Spot Localization of Nonlinearity" *Symmetry* 10, no. 5: 156.
https://doi.org/10.3390/sym10050156