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Kumar S, Li P, Zeng L, He J, Malomed BA. A solvable model for symmetry-breaking phase transitions. Sci Rep 2023; 13:13768. [PMID: 37612417 PMCID: PMC10447515 DOI: 10.1038/s41598-023-40704-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2023] [Accepted: 08/16/2023] [Indexed: 08/25/2023] Open
Abstract
Analytically solvable models are benchmarks in studies of phase transitions and pattern-forming bifurcations. Such models are known for phase transitions of the second kind in uniform media, but not for localized states (solitons), as integrable equations which produce solitons do not admit intrinsic transitions in them. We introduce a solvable model for symmetry-breaking phase transitions of both the first and second kinds (alias sub- and supercritical bifurcations) for solitons pinned to a combined linear-nonlinear double-well potential, represented by a symmetric pair of delta-functions. Both self-focusing and defocusing signs of the nonlinearity are considered. In the former case, exact solutions are produced for symmetric and asymmetric solitons. The solutions explicitly demonstrate a switch between the symmetry-breaking transitions of the first and second kinds (i.e., sub- and supercritical bifurcations, respectively). In the self-defocusing model, the solution demonstrates the transition of the second kind which breaks antisymmetry of the first excited state.
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Affiliation(s)
- Shatrughna Kumar
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv, 69978, Israel
| | - Pengfei Li
- Department of Physics, Taiyuan Normal University, Jinzhong, 030619, China
| | - Liangwei Zeng
- Department of Basic Course, Guangzhou Maritime University, Guangzhou, 510725, China
| | - Jingsong He
- Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv, 69978, Israel.
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile.
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Kumar S, Li P, Malomed BA. One-dimensional Townes solitons in dual-core systems with localized coupling. Phys Rev E 2023; 108:024202. [PMID: 37723768 DOI: 10.1103/physreve.108.024202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2023] [Accepted: 07/06/2023] [Indexed: 09/20/2023]
Abstract
The recent creation of Townes solitons (TSs) in binary Bose-Einstein condensates and experimental demonstration of spontaneous symmetry breaking (SSB) in solitons propagating in dual-core optical fibers has drawn renewed interest in the TS and SSB phenomenology in these and other settings. In particular, stabilization of TSs, which are always unstable in free space, is a relevant problem with various ramifications. We introduce a system which admits exact solutions for both TSs and SSB of solitons. It is based on a dual-core waveguide with quintic self-focusing and fused (localized) coupling between the cores. The respective system of coupled nonlinear Schrödinger equations gives rise to exact solutions for full families of symmetric and asymmetric solitons, which are produced by the supercritical SSB bifurcation (i.e., the symmetry-breaking phase transition of the second kind). Stability boundaries of asymmetric solitons are identified by dint of numerical methods. Unstable solitons spontaneously transform into robust moderately asymmetric breathers or strongly asymmetric states with small intrinsic oscillations. The setup can be used in the design of photonic devices operating in coupling and switching regimes.
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Affiliation(s)
- Shatrughna Kumar
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
| | - Pengfei Li
- Department of Physics, Taiyuan Normal University, Jinzhong, 030619, China
- Institute of Computational and Applied Physics, Taiyuan Normal University, Jinzhong, 030619, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
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Yan H, Wu G, Ren J. Reciprocal conditions in one-dimensional nonlinear wave systems. Phys Rev E 2019; 100:012207. [PMID: 31499783 DOI: 10.1103/physreve.100.012207] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2019] [Indexed: 11/07/2022]
Abstract
We study wave reciprocity in one-dimensional asymmetric systems constructed by multiple nonlinear δ-function scatters embedded within linear scatters. A general reciprocal condition is proposed, in terms of the rotation symmetry between forward and backward transfer matrices. We then derive various resonance conditions, under which all scatterers behave as merging into either a single nonlinear δ scatter, or a symmetric nonlinear barrier configuration. As such, the reciprocity appears periodically by changing widths of linear constant potentials between neighboring nonlinear δ scatters. Moreover, the wave reciprocity will not be violated if one replaces the linear constant potential between two δ-nonlinear scatters with any other kind of transparent scatterers.
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Affiliation(s)
- Hengzhe Yan
- Center for Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Sciences and Engineering, Tongji University, 200092 Shanghai, China
| | - Gaomin Wu
- Center for Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Sciences and Engineering, Tongji University, 200092 Shanghai, China
| | - Jie Ren
- Center for Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Sciences and Engineering, Tongji University, 200092 Shanghai, China
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Doelman A, van Heijster P, Shen J. Pulse dynamics in reaction-diffusion equations with strong spatially localized impurities. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:rsta.2017.0183. [PMID: 29507168 PMCID: PMC5869605 DOI: 10.1098/rsta.2017.0183] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 09/27/2017] [Indexed: 05/03/2023]
Abstract
In this article, a general geometric singular perturbation framework is developed to study the impact of strong, spatially localized, nonlinear impurities on the existence, stability and bifurcations of localized structures in systems of linear reaction-diffusion equations. By taking advantage of the multiple-scale nature of the problem, we derive algebraic conditions determining the existence and stability of pinned single- and multi-pulse solutions. Our methods enable us to explicitly control the spectrum associated with a (multi-)pulse solution. In the scalar case, we show how eigenvalues may move in and out of the essential spectrum and that Hopf bifurcations cannot occur. By contrast, even a pinned 1-pulse solution can undergo a Hopf bifurcation in a two-component system of linear reaction-diffusion equations with (only) one impurity.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.
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Affiliation(s)
- Arjen Doelman
- Mathematisch Instituut, Leiden University, 2300 RA Leiden, The Netherlands
| | - Peter van Heijster
- School of Mathematical Sciences, Queensland University of Technology, Brisbane, Queensland 4000, Australia
| | - Jianhe Shen
- School of Mathematics and Computer Science and FJKLMAA, Fujian Normal University, Fuzhou 350117, People Republic of China
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Lutsky V, Malomed BA. Solitons supported by singular modulation of the cubic nonlinearity. OPTICS EXPRESS 2017; 25:12967-12983. [PMID: 28786648 DOI: 10.1364/oe.25.012967] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2017] [Accepted: 04/11/2017] [Indexed: 06/07/2023]
Abstract
A model of the optical media with a spatially structured Kerr nonlinearity is introduced. The nonlinearity strength is modulated by a set of singular peaks on top of a self-focusing or defocusing uniform background. The peaks may include a repulsive or attractive linear potential too. We find that a pair of mutually symmetric peaks readily gives rise to the spontaneous symmetry breaking (SSB) of modes pinned to individual peaks, while antisymmetric pinned modes are always unstable, transforming into robust spatially odd breathers. Three- and five-peak structures support symmetric modes, with in-phase or twisted profiles, and do not give rise to asymmetric states. A stability area is found for the twisted states pinned to the triple peaks, while the corresponding in-phase modes are unstable, unless the three modulation peaks are set very close to each other, covered by a single-peak pinned mode. All patterns pinned to five peaks are unstable too. Collisions of moving solitons with the singular-modulation peak are studied too. Slowly moving solitons bounce back from the peak, while the collisions are quasi-elastic for fast solitons. In the intermediate case, the soliton is destroyed by the collision. In a special case, the condition of a resonance of the incident soliton with a trapped mode supported by the peak leads to capture of the soliton.
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Lepri S, Pikovsky A. Nonreciprocal wave scattering on nonlinear string-coupled oscillators. CHAOS (WOODBURY, N.Y.) 2014; 24:043119. [PMID: 25554039 DOI: 10.1063/1.4899205] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We study scattering of a periodic wave in a string on two lumped oscillators attached to it. The equations can be represented as a driven (by the incident wave) dissipative (due to radiation losses) system of delay differential equations of neutral type. Nonlinearity of oscillators makes the scattering non-reciprocal: The same wave is transmitted differently in two directions. Periodic regimes of scattering are analyzed approximately, using amplitude equation approach. We show that this setup can act as a nonreciprocal modulator via Hopf bifurcations of the steady solutions. Numerical simulations of the full system reveal nontrivial regimes of quasiperiodic and chaotic scattering. Moreover, a regime of a "chaotic diode," where transmission is periodic in one direction and chaotic in the opposite one, is reported.
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Affiliation(s)
- Stefano Lepri
- Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
| | - Arkady Pikovsky
- Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str 24/25, Potsdam, Germany
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D'Ambroise J, Kevrekidis PG, Lepri S. Eigenstates and instabilities of chains with embedded defects. CHAOS (WOODBURY, N.Y.) 2013; 23:023109. [PMID: 23822474 DOI: 10.1063/1.4803523] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
We consider the eigenvalue problem for one-dimensional linear Schrödinger lattices (tight-binding) with an embedded few-sites linear or nonlinear, Hamiltonian or non-conservative defect (an oligomer). Such a problem arises when considering scattering states in the presence of (generally complex) impurities as well as in the stability analysis of nonlinear waves. We describe a general approach based on a matching of solutions of the linear portions of the lattice at the location of the oligomer defect. As specific examples, we discuss both linear and nonlinear, Hamiltonian and PT-symmetric dimers and trimers. In the linear case, this approach provides us a handle for semi-analytically computing the spectrum [this amounts to the solution of a polynomial equation]. In the nonlinear case, it enables the computation of the linearization spectrum around the stationary solutions. The calculations showcase the oscillatory instabilities that strongly nonlinear states typically manifest.
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Affiliation(s)
- J D'Ambroise
- Department of Mathematics, Bard College, Annandale-on-Hudson, New York 12504, USA
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Tsang CH, Malomed BA, Chow KW. Multistable dissipative structures pinned to dual hot spots. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:066609. [PMID: 22304213 DOI: 10.1103/physreve.84.066609] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2011] [Revised: 10/07/2011] [Indexed: 05/31/2023]
Abstract
We analyze the formation of one-dimensional localized patterns in a nonlinear dissipative medium including a set of two narrow "hot spots" (HSs), which carry the linear gain, local potential, cubic self-interaction, and cubic loss, while the linear loss acts in the host medium. This system can be realized as a spatial-domain one in optics and also in Bose-Einstein condensates of quasiparticles in solid-state settings. Recently, exact solutions were found for localized modes pinned to the single HS represented by the δ function. The present paper reports analytical and numerical solutions for coexisting two- and multipeak modes, which may be symmetric or antisymmetric with respect to the underlying HS pair. Stability of the modes is explored through simulations of their perturbed evolution. The sign of the cubic nonlinearity plays a crucial role: in the case of the self-focusing, only the fundamental symmetric and antisymmetric modes, with two local peaks tacked to the HSs, and no additional peaks between them, may be stable. In this case, all the higher-order multipeak modes, being unstable, evolve into the fundamental ones. Stability regions for the fundamental modes are reported. A more interesting situation is found in the case of the self-defocusing cubic nonlinearity, with the HS pair giving rise to a multistability, with up to eight coexisting stable multipeak patterns, symmetric and antisymmetric ones. The system without the self-interaction, the nonlinearity being represented only by the local cubic loss, is investigated too. This case is similar to those with the self-focusing or defocusing nonlinearity, if the linear potential of the HS is, respectively, attractive or repulsive. An additional feature of the former setting is the coexistence of the stable fundamental modes with robust breathers.
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Affiliation(s)
- Cheng Hou Tsang
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong
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Li K, Kevrekidis PG, Malomed BA, Frantzeskakis DJ. Transfer and scattering of wave packets by a nonlinear trap. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:056609. [PMID: 22181538 DOI: 10.1103/physreve.84.056609] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2011] [Revised: 09/18/2011] [Indexed: 05/31/2023]
Abstract
In the framework of a one-dimensional model with a tightly localized self-attractive nonlinearity, we study the formation and transfer (dragging) of a trapped mode by "nonlinear tweezers," as well as the scattering of coherent linear wave packets on the stationary localized nonlinearity. The use of a nonlinear trap for dragging allows one to pick up and transfer the relevant structures without grabbing surrounding "radiation." A stability border for the dragged modes is identified by means of analytical estimates and systematic simulations. In the framework of the scattering problem, the shares of trapped, reflected, and transmitted wave fields are found. Quasi-Airy stationary modes with a divergent norm, which may be dragged by a nonlinear trap moving at a constant acceleration, are briefly considered too.
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Affiliation(s)
- Kai Li
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-9305, USA
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Miroshnichenko AE, Molina MI, Kivshar YS. Localized modes and bistable scattering in nonlinear network junctions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:046602. [PMID: 17501002 DOI: 10.1103/physreve.75.046602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/19/2006] [Indexed: 05/15/2023]
Abstract
We study the properties of junctions created by the crossing of N identical branches of linear discrete networks. We reveal that for N>2 such a junction creates a topological defect and supports two types of spatially localized modes. We analyze the wave scattering by the junction defect and demonstrate nonzero reflection for any set of parameters. If the junction is nonlinear, it is possible to achieve the maximum transmission for any frequency by tuning the intensity of the scattering wave. In addition, near the maximum transmission the system shows the bistable behavior.
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Affiliation(s)
- Andrey E Miroshnichenko
- Nonlinear Physics Centre, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 0200, Australia
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Miroshnichenko AE, Flach S, Malomed B. Resonant scattering of solitons. CHAOS (WOODBURY, N.Y.) 2003; 13:874-879. [PMID: 12946179 DOI: 10.1063/1.1597071] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We study the scattering of solitons in the nonlinear Schrödinger equation on local inhomogeneities which may give rise to resonant transmission and reflection. In both cases, we derive resonance conditions for the soliton's velocity. The analytical predictions are tested numerically in regimes characterized by various time scales. Special attention is paid to intermode interactions and their effect on coherence, decoherence, and dephasing of plane-wave modes which build up the soliton.
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Affiliation(s)
- A E Miroshnichenko
- Max-Planck-Institut fur Physik komplexer Systeme, Nothnitzer Strasse 38, D-01187 Dresden, Germany
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Bulashenko OM, Kochelap VA, Bonilla LL. Coherent patterns and self-induced diffraction of electrons on a thin nonlinear layer. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:1537-1540. [PMID: 9985992 DOI: 10.1103/physrevb.54.1537] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Diagrammatic treatment of light scattering by a collection of Kerr particles. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 50:2921-2940. [PMID: 9976534 DOI: 10.1103/physrevb.50.2921] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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