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Yin HM, Li JH, Zheng Z, Chiang KS, Chow KW. Robustness and stability of doubly periodic patterns of the focusing nonlinear Schrödinger equation. CHAOS (WOODBURY, N.Y.) 2024; 34:013120. [PMID: 38231179 DOI: 10.1063/5.0157966] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/13/2023] [Accepted: 12/15/2023] [Indexed: 01/18/2024]
Abstract
The nonlinear Schrödinger equation possesses doubly periodic solutions expressible in terms of the Jacobi elliptic functions. Such solutions can be realized through doubly periodic patterns observed in experiments in fluid mechanics and optics. Stability and robustness of these doubly periodic wave profiles in the focusing regime are studied computationally by using two approaches. First, linear stability is considered by Floquet theory. Growth will occur if the eigenvalues of the monodromy matrix are of a modulus larger than unity. This is verified by numerical simulations with input patterns of different periods. Initial patterns associated with larger eigenvalues will disintegrate faster due to instability. Second, formation of these doubly periodic patterns from a tranquil background is scrutinized. Doubly periodic profiles are generated by perturbing a continuous wave with one Fourier mode, with or without the additional presence of random noise. Effects of varying phase difference, perturbation amplitude, and randomness are studied. Varying the phase angle has a dramatic influence. Periodic patterns will only emerge if the perturbation amplitude is not too weak. The growth of higher-order harmonics, as well as the formation of breathers and repeating patterns, serve as a manifestation of the classical problem of Fermi-Pasta-Ulam-Tsingou recurrence.
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Affiliation(s)
- H M Yin
- Department of Mechanical Engineering, University of Hong Kong, Hong Kong, China
| | - J H Li
- School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
| | - Z Zheng
- School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
| | - K S Chiang
- Department of Electrical Engineering, City University of Hong Kong, Hong Kong, China
| | - K W Chow
- Department of Mechanical Engineering, University of Hong Kong, Hong Kong, China
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Vanderhaegen G, Naveau C, Szriftgiser P, Kudlinski A, Conforti M, Mussot A, Onorato M, Trillo S, Chabchoub A, Akhmediev N. "Extraordinary" modulation instability in optics and hydrodynamics. Proc Natl Acad Sci U S A 2021; 118:e2019348118. [PMID: 33790009 PMCID: PMC8040794 DOI: 10.1073/pnas.2019348118] [Citation(s) in RCA: 15] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The classical theory of modulation instability (MI) attributed to Bespalov-Talanov in optics and Benjamin-Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.
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Affiliation(s)
- Guillaume Vanderhaegen
- University of Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France;
| | - Corentin Naveau
- University of Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - Pascal Szriftgiser
- University of Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - Alexandre Kudlinski
- University of Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - Matteo Conforti
- University of Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - Arnaud Mussot
- University of Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
- Institut Universitaire de France, F-75005 Paris, France
| | - Miguel Onorato
- Dipartimento di Fisica, Università degli Studi di Torino, 10125 Torino, Italy
| | - Stefano Trillo
- Department of Engineering, University of Ferrara, 44122 Ferrara, Italy
| | - Amin Chabchoub
- Centre for Wind, Waves and Water, School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia
- Disaster Prevention Research Institute, Kyoto University, Kyoto 611-0011, Japan
| | - Nail Akhmediev
- Department of Theoretical Physics, Research School of Physics, The Australian National University, Canberra, ACT 2600, Australia
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He XG, Zhao D, Li L, Luo HG. Engineering integrable nonautonomous nonlinear Schrödinger equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:056610. [PMID: 19518585 DOI: 10.1103/physreve.79.056610] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2009] [Revised: 05/01/2009] [Indexed: 05/27/2023]
Abstract
We investigate Painlevé integrability of a generalized nonautonomous one-dimensional nonlinear Schrödinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painlevé analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painlevé integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.
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Affiliation(s)
- Xu-Gang He
- School of Mathematics and Statistics, Center for Interdisciplinary Studies, Department of Modern Physics, Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, China
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Petnikova VM, Shuvalov VV. Multicomponent cnoidal waves in cascade quasisynchronous frequency conversion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:026605. [PMID: 19391861 DOI: 10.1103/physreve.79.026605] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/04/2008] [Indexed: 05/27/2023]
Abstract
It is shown that cascade quasisynchronous frequency conversion due to quadratic nonlinearity can be described in terms of an effective cubic nonlinearity. This enables one to reduce a four-mode interaction problem to solving a system of two coupled nonlinear Schrödinger equations for the amplitudes of the waves participating in both nonlinear processes. Exact analytic solutions of the corresponding system are found in the form of multicomponent cnoidal waves with components expressed through a sum and a difference of two similar fundamental solutions of the Lamé equation with shifted arguments. It is shown that solutions obtained in such a way enable one to optimize the conversion efficiency because of full coverage of the range of possible boundary conditions.
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Affiliation(s)
- V M Petnikova
- M.V. Lomonosov Moscow State University, Vorob'evy Gory, Moscow 119992, Russia
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Wang L, Li L, Li Z, Zhou G, Mihalache D. Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:036614. [PMID: 16241599 DOI: 10.1103/physreve.72.036614] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2005] [Indexed: 05/05/2023]
Abstract
The generalized nonlinear Schrödinger model with distributed dispersion, nonlinearity, and gain or loss is considered and the explicit, analytical solutions describing the dynamics of bright solitons on a continuous-wave background are obtained in quadratures. Then, the generation, compression, and propagation of pulse trains are discussed in detail. The numerical results show that solitons can be compressed by choosing the appropriate control fiber system, and pulse trains generated by modulation instability can propagate undistorsted along fibers with distributed parameters by controlling appropriately the energy of each pulse in the pulse train.
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Affiliation(s)
- Luyun Wang
- College of Physics and Electronics Engineering, and Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
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Chow KW, Lai DWC. Periodic solutions for systems of coupled nonlinear Schrödinger equations with three and four components. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:017601. [PMID: 12935291 DOI: 10.1103/physreve.68.017601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2003] [Indexed: 05/24/2023]
Abstract
Periodic solutions for systems of coupled nonlinear Schrödinger equations (CNLS) are established by the Hirota bilinear method and elliptic functions. The interesting feature is the choice of theta functions in the formulation. The sum of moduli of the components or the total intensity of the beam in physical terms, will now be a rational function, instead of a polynomial, of elliptic functions. Each component of the CNLS may have multiple peaks within one period.
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Affiliation(s)
- K W Chow
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong.
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Mihalache D, Torner L, Moldoveanu F, Panoiu NC, Truta N. Soliton solutions for a perturbed nonlinear Schrodinger equation. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/26/17/001] [Citation(s) in RCA: 43] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Mihalache D, Panoiu NC, Moldoveanu F, Baboiu DM. The Riemann problem method for solving a perturbed nonlinear Schrodinger equation describing pulse propagation in optic fibres. ACTA ACUST UNITED AC 1999. [DOI: 10.1088/0305-4470/27/18/026] [Citation(s) in RCA: 55] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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Burak D. Steering of bright-soliton pairs excited by symmetric and real initial profiles. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1995; 52:4054-4058. [PMID: 9912719 DOI: 10.1103/physreva.52.4054] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Panoiu N, Mihalache D, Baboiu D. Dark-soliton timing jitter caused by fluctuations in initial pulse shape. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1995; 52:4182-4186. [PMID: 9912736 DOI: 10.1103/physreva.52.4182] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Mihalache D, Torner L, Moldoveanu F, Panoiu N, Truta N. Inverse-scattering approach to femtosecond solitons in monomode optical fibers. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 48:4699-4709. [PMID: 9961154 DOI: 10.1103/physreve.48.4699] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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