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Bai XD, Xu T, Zhao Y, Li T, Ji G, Zhao J. Defect-induced discrete breather in dissipative optical lattices with weak nonlinearity. OPTICS EXPRESS 2024; 32:20503-20514. [PMID: 38859431 DOI: 10.1364/oe.522409] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2024] [Accepted: 05/11/2024] [Indexed: 06/12/2024]
Abstract
It is widely believed that the discrete breather (DB) can only be created when the nonlinearity is strong in nonlinear systems. However, we here establish that this belief is incorrect. In this work, we systemically investigate the generation of DBs induced by coupling of the defects and nonlinearity for Bose-Einstein condensates in dissipative optical lattices. The results show that, only in a clean lattice is strong nonlinearity a necessary condition for generating of DB; whereas, if the lattice has a defect, the DBs can also be discovered even in weak nonlinearity, and its generation turns out to be controllable. In addition, we further reveal a critical interval of the defect in weak nonlinearity, within which DBs can be found, while outside DBs do not exist. Furthermore, we also explore the impact of multiple defects on the generation of DBs, and analyze the underlying physical mechanisms of these interesting phenomena. The results not only have the potential to be used for more precise engineering in the DB experiments, but also suggest that the DB may be ubiquitous since the defects and dissipation are unavoidable in real physics.
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Thomson SJ, Durey M, Rosales RR. Discrete and periodic complex Ginzburg-Landau equation for a hydrodynamic active lattice. Phys Rev E 2021; 103:062215. [PMID: 34271671 DOI: 10.1103/physreve.103.062215] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2020] [Accepted: 05/30/2021] [Indexed: 11/07/2022]
Abstract
A discrete and periodic complex Ginzburg-Landau equation, coupled to a mean equation, is systematically derived from a driven and dissipative lattice oscillator model, close to the onset of a supercritical Andronov-Hopf bifurcation. The oscillator model is inspired by recent experiments exploring active vibrations of quasi-one-dimensional lattices of self-propelled millimetric droplets bouncing on a vertically vibrating fluid bath. Our systematic derivation provides a direct link between the constitutive properties of the lattice system and the coefficients of the resultant amplitude equations, paving the way to compare the emergent nonlinear dynamics-namely, the onset and formation of discrete dark solitons, breathers, and traveling waves-against experiments. The framework presented herein is expected to be applicable to a wider class of oscillators characterized by the presence of a dynamic coupling potential between particles. More broadly, our results point to deeper connections between nonlinear oscillators and the physics of active and driven matter.
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Affiliation(s)
- Stuart J Thomson
- School of Engineering, Brown University, Providence, Rhode Island 02912, USA.,Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Matthew Durey
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Rodolfo R Rosales
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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Susanto H, Kusdiantara R, Li N, Kirikchi OB, Adzkiya D, Putri ERM, Asfihani T. Snakes and ghosts in a parity-time-symmetric chain of dimers. Phys Rev E 2018; 97:062204. [PMID: 30011512 DOI: 10.1103/physreve.97.062204] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2017] [Indexed: 11/07/2022]
Abstract
We consider linearly coupled discrete nonlinear Schrödinger equations with gain and loss terms and with a cubic-quintic nonlinearity. The system models a parity-time (PT)-symmetric coupler composed by a chain of dimers. We study uniform states and site-centered and bond-centered spatially localized solutions and present that each solution has a symmetric and antisymmetric configuration between the arms. The symmetric solutions can become unstable due to bifurcations of asymmetric ones, that are called ghost states, because they exist only when an otherwise real propagation constant is taken to be complex valued. When a parameter is varied, the resulting bifurcation diagrams for the existence of standing localized solutions have a snaking behavior. The critical gain and loss coefficient above which the PT symmetry is broken corresponds to the condition when bifurcation diagrams of symmetric and antisymmetric states merge. Past the symmetry breaking, the system no longer has time-independent states. Nevertheless, equilibrium solutions can be analytically continued by defining a dual equation that leads to ghost states associated with growth or decay, that are also identified and examined here. We show that ghost localized states also exhibit snaking bifurcation diagrams. We analyze the width of the snaking region and provide asymptotic approximations in the limit of strong and weak coupling where good agreement is obtained.
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Affiliation(s)
- H Susanto
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - R Kusdiantara
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom.,Centre of Mathematical Modelling and Simulation, Institut Teknologi Bandung, 1st Floor, Labtek III, Jl. Ganesha No. 10, Bandung, 40132, Indonesia
| | - N Li
- School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - O B Kirikchi
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
| | - D Adzkiya
- Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Sukolilo, Surabaya 60111, Indonesia
| | - E R M Putri
- Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Sukolilo, Surabaya 60111, Indonesia
| | - T Asfihani
- Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Sepuluh Nopember, Sukolilo, Surabaya 60111, Indonesia
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Kusdiantara R, Susanto H. Homoclinic snaking in the discrete Swift-Hohenberg equation. Phys Rev E 2017; 96:062214. [PMID: 29347380 DOI: 10.1103/physreve.96.062214] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2016] [Indexed: 11/07/2022]
Abstract
We consider the discrete Swift-Hohenberg equation with cubic and quintic nonlinearity, obtained from discretizing the spatial derivatives of the Swift-Hohenberg equation using central finite differences. We investigate the discretization effect on the bifurcation behavior, where we identify three regions of the coupling parameter, i.e., strong, weak, and intermediate coupling. Within the regions, the discrete Swift-Hohenberg equation behaves either similarly or differently from the continuum limit. In the intermediate coupling region, multiple Maxwell points can occur for the periodic solutions and may cause irregular snaking and isolas. Numerical continuation is used to obtain and analyze localized and periodic solutions for each case. Theoretical analysis for the snaking and stability of the corresponding solutions is provided in the weak coupling region.
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Affiliation(s)
- R Kusdiantara
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom.,Centre of Mathematical Modelling and Simulation, Institut Teknologi Bandung, 1st Floor, Labtek III, Jl. Ganesha No. 10, Bandung 40132, Indonesia
| | - H Susanto
- Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom
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Bai XD, Malomed BA, Deng FG. Unidirectional transport of wave packets through tilted discrete breathers in nonlinear lattices with asymmetric defects. Phys Rev E 2016; 94:032216. [PMID: 27739723 DOI: 10.1103/physreve.94.032216] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2016] [Indexed: 11/07/2022]
Abstract
We consider the transfer of lattice wave packets through a tilted discrete breather (TDB) in opposite directions in the discrete nonlinear Schrödinger model with asymmetric defects, which may be realized as a Bose-Einstein condensate trapped in a deep optical lattice, or as optical beams in a waveguide array. A unidirectional transport mode is found, in which the incident wave packets, whose energy belongs to a certain interval between full reflection and full passage regions, pass the TDB only in one direction, while in the absence of the TDB, the same lattice admits bidirectional propagation. The operation of this mode is accurately explained by an analytical consideration of the respective energy barriers. The results suggest that the TDB may emulate the unidirectional propagation of atomic and optical beams in various settings. In the case of the passage of the incident wave packet, the scattering TDB typically shifts by one lattice unit in the direction from which the wave packet arrives, which is an example of the tractor-beam effect, provided by the same system, in addition to the rectification of incident waves.
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Affiliation(s)
- Xiao-Dong Bai
- Department of Physics and Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel.,Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101, Russia
| | - Fu-Guo Deng
- Department of Physics and Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China
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Bai XD, Zhang M, Xiong J, Yang GJ, Deng FG. Selective distillation phenomenon in two-species Bose-Einstein condensates in open boundary optical lattices. Sci Rep 2015; 5:17101. [PMID: 26597592 PMCID: PMC4656982 DOI: 10.1038/srep17101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/04/2015] [Accepted: 10/26/2015] [Indexed: 11/30/2022] Open
Abstract
We investigate the formation of discrete breathers (DBs) and the dynamics of the mixture of two-species Bose-Einstein condensates (BECs) in open boundary optical lattices using the discrete nonlinear Schrödinger equations. The results show that the coupling of intra- and interspecies interaction can lead to the existence of pure single-species DBs and symbiotic DBs (i.e., two-species DBs). Furthermore, we find that there is a selective distillation phenomenon in the dynamics of the mixture of two-species BECs. One can selectively distil one species from the mixture of two-species BECs and can even control dominant species fraction by adjusting the intra- and interspecies interaction in optical lattices. Our selective distillation mechanism may find potential application in quantum information storage and quantum information processing based on multi-species atoms.
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Affiliation(s)
- Xiao-Dong Bai
- Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China
| | - Mei Zhang
- Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China
| | - Jun Xiong
- Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China
| | - Guo-Jian Yang
- Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China
| | - Fu-Guo Deng
- Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China
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Archilla JFR, Kosevich YA, Jiménez N, Sánchez-Morcillo VJ, García-Raffi LM. Ultradiscrete kinks with supersonic speed in a layered crystal with realistic potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:022912. [PMID: 25768574 DOI: 10.1103/physreve.91.022912] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2014] [Indexed: 06/04/2023]
Abstract
In this paper we develop a dynamical model of the propagating nonlinear localized excitations, supersonic kinks, in the cation layer in a silicate mica crystal. We start from purely electrostatic Coulomb interaction and add the Ziegler-Biersack-Littmark short-range repulsive potential and the periodic potential produced by other atoms of the lattice. The proposed approach allows the construction of supersonic kinks which can propagate in the lattice within a large range of energies and velocities. Due to the presence of the short-range repulsive component in the potential, the interparticle distances in the lattice kinks with high energy are limited by physically reasonable values. The introduction of the periodic lattice potential results in the important feature that the kinks propagate with the single velocity and single energy, which are independent on the excitation conditions. The unique average velocity of the supersonic kinks on the periodic substrate potential we relate with the kink amplitude of the relative particle displacements, which is determined by the interatomic distance corresponding to the minimum of the total, interparticle plus substrate, lattice potential. The found kinks are ultradiscrete and can be described with the "magic wave number" q=2π/3a, which was previously revealed in the nonlinear sinusoidal waves and supersonic kinks in the Fermi-Pasta-Ulam lattice. The extreme discreteness of the observed supersonic kinks, with basically two particles moving at the same time, allows the detailed interpretation of their double-kink structure, which is not possible for the multikinks without an account for the lattice discreteness. Analytical calculations of the displacement patterns and energies of the supersonic kinks are confirmed by numerical simulations. The computed energy of the found supersonic kinks in the considered realistic lattice potential is in a good agreement with the experimental evidence for the transport of localized energetic excitations in silicate mica crystals between the points of ^{40}K recoil and subsequent sputtering.
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Affiliation(s)
- J F R Archilla
- Grupo de Física No Lineal, Universidad de Sevilla, ETSI Informática, Avda Reina Mercedes s/n, 41012 Seville, Spain
| | - Yu A Kosevich
- Semenov Institute of Chemical Physics, Russian Academy of Sciences. Kosygin street 4, 119991 Moscow, Russia
| | - N Jiménez
- Instituto de Investigación para la Gestión, Integrada de las Zonas Costeras, Universidad Politécnica de Valencia, C/.Paranimfo 1, 46730 Grao de Gandia, Spain
| | - V J Sánchez-Morcillo
- Instituto de Investigación para la Gestión, Integrada de las Zonas Costeras, Universidad Politécnica de Valencia, C/.Paranimfo 1, 46730 Grao de Gandia, Spain
| | - L M García-Raffi
- Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera s/n, 46022 Valencia, Spain
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Bai XD, Zhang AX, Xue JK. Transfer of dipolar gas through the discrete localized mode. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062916. [PMID: 24483540 DOI: 10.1103/physreve.88.062916] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2013] [Indexed: 06/03/2023]
Abstract
By considering the discrete nonlinear Schrödinger model with dipole-dipole interactions for dipolar condensate, the existence, the types, the stability, and the dynamics of the localized modes in a nonlinear lattice are discussed. It is found that the contact interaction and the dipole-dipole interactions play important roles in determining the existence, the type, and the stability of the localized modes. Because of the coupled effects of the contact interaction and the dipole-dipole interactions, rich localized modes and their stability nature can exist: when the contact interaction is larger and the dipole-dipole interactions is smaller, a discrete bright breather occurs. In this case, while the on-site interaction can stabilize the discrete breather, the dipole-dipole interactions will destabilize the discrete breather; when both the contact interaction and the dipole-dipole interactions are larger, a discrete kink appears. In this case, both the on-site interaction and the dipole-dipole interactions can stabilize the discrete kink, but the discrete kink is more unstable than the ordinary discrete breather. The predicted results provide a deep insight into the dynamics of blocking, filtering, and transfer of the norm in nonlinear lattices for dipolar condensates.
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Affiliation(s)
- Xiao-Dong Bai
- Key Laboratory of Atomic & Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronics Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Ai-Xia Zhang
- Key Laboratory of Atomic & Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronics Engineering, Northwest Normal University, Lanzhou 730070, China
| | - Ju-Kui Xue
- Key Laboratory of Atomic & Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronics Engineering, Northwest Normal University, Lanzhou 730070, China
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Mejía-Cortés C, Vicencio RA, Malomed BA. Mobility of solitons in one-dimensional lattices with the cubic-quintic nonlinearity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052901. [PMID: 24329325 DOI: 10.1103/physreve.88.052901] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2013] [Indexed: 06/03/2023]
Abstract
We investigate mobility regimes for localized modes in the discrete nonlinear Schrödinger (DNLS) equation with the cubic-quintic on-site terms. Using the variational approximation, the largest soliton's total power admitting progressive motion of kicked discrete solitons is predicted by comparing the effective kinetic energy with the respective Peierls-Nabarro (PN) potential barrier. The prediction, for the DNLS model with the cubic-only nonlinearity too, demonstrates a reasonable agreement with numerical findings. A small self-focusing quintic term quickly suppresses the mobility. In the case of the competition between the cubic self-focusing and quintic self-defocusing terms, we identify parameter regions where odd and even fundamental modes exchange their stability, involving intermediate asymmetric modes. In this case, stable solitons can be set in motion by kicking, so as to let them pass the PN barrier. Unstable solitons spontaneously start oscillatory or progressive motion, if they are located, respectively, below or above a mobility threshold. Collisions between moving discrete solitons, at the competing nonlinearities frame, are studied too.
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Affiliation(s)
- C Mejía-Cortés
- Departamento de Física and MSI-Nucleus on Advanced Optics, Center for Optics and Photonics (CEFOP), Facultad de Ciencias, Universidad de Chile, Santiago, Chile
| | - Rodrigo A Vicencio
- Departamento de Física and MSI-Nucleus on Advanced Optics, Center for Optics and Photonics (CEFOP), Facultad de Ciencias, Universidad de Chile, Santiago, Chile
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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Dawes JHP, Susanto H. Variational approximation and the use of collective coordinates. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:063202. [PMID: 23848797 DOI: 10.1103/physreve.87.063202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2012] [Revised: 04/02/2013] [Indexed: 06/02/2023]
Abstract
We consider propagating, spatially localized waves in a class of equations that contain variational and nonvariational terms. The dynamics of the waves is analyzed through a collective coordinate approach. Motivated by the variational approximation, we show that there is a natural choice of projection onto collective variables for reducing the governing (nonlinear) partial differential equation (PDE) to coupled ordinary differential equations (ODEs). This projection produces ODEs whose solutions are exactly the stationary states of the effective Lagrangian that would be considered in applying the variational approximation method. We illustrate our approach by applying it to a modified Fisher equation for a traveling front, containing a non-constant-coefficient nonlinear term. We present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the PDE much more closely than previously proposed projections.
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Affiliation(s)
- J H P Dawes
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
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