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Colcelli A, Mussardo G, Sierra G, Trombettoni A. Integrable Floquet Hamiltonian for a Periodically Tilted 1D Gas. PHYSICAL REVIEW LETTERS 2019; 123:130401. [PMID: 31697557 DOI: 10.1103/physrevlett.123.130401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2019] [Revised: 06/19/2019] [Indexed: 06/10/2023]
Abstract
An integrable model subjected to a periodic driving gives rise generally to a nonintegrable Floquet Hamiltonian. Here we show that the Floquet Hamiltonian of the integrable Lieb-Liniger model in the presence of a linear potential with a periodic time-dependent strength is instead integrable and its quasienergies can be determined using the Bethe ansatz approach. We discuss various aspects of the dynamics of the system at stroboscopic times and we also propose a possible experimental realization of the periodically driven tilting in terms of a shaken rotated ring potential.
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Affiliation(s)
- A Colcelli
- SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136 Trieste, Italy
| | - G Mussardo
- SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136 Trieste, Italy
| | - G Sierra
- Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid, 28049 Madrid, Spain
| | - A Trombettoni
- SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136 Trieste, Italy
- CNR-IOM DEMOCRITOS Simulation Center, Via Bonomea 265, I-34136 Trieste, Italy
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2
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Conforti M, Mussot A, Kudlinski A, Rota Nodari S, Dujardin G, De Biévre S, Armaroli A, Trillo S. Heteroclinic Structure of Parametric Resonance in the Nonlinear Schrödinger Equation. PHYSICAL REVIEW LETTERS 2016; 117:013901. [PMID: 27419569 DOI: 10.1103/physrevlett.117.013901] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/18/2016] [Indexed: 06/06/2023]
Abstract
We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearized Floquet analysis.
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Affiliation(s)
- M Conforti
- Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - A Mussot
- Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - A Kudlinski
- Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
| | - S Rota Nodari
- IMB UMR 5584, CNRS, Université Bourgogne Franche-Comté, F-21000 Dijon, France
| | - G Dujardin
- Univ. Lille, CNRS, UMR 8524-Laboratoire Paul Painlevé; Equipe MEPHYSTO, INRIA, F-59000 Lille, France
| | - S De Biévre
- Univ. Lille, CNRS, UMR 8524-Laboratoire Paul Painlevé; Equipe MEPHYSTO, INRIA, F-59000 Lille, France
| | - A Armaroli
- FOTON (CNRS-UMR 6082) Université de Rennes 1, ENSSAT, 6 rue de Kerampont, CS 80518, F-22305 Lannion Cedex, France
| | - S Trillo
- Dipartimento di Ingegneria, Università di Ferrara, Via Saragat 1, 44122 Ferrara, Italy
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Quintero NR, Mertens FG, Bishop AR. Soliton stability criterion for generalized nonlinear Schrödinger equations. Phys Rev E 2015; 91:012905. [PMID: 25679680 DOI: 10.1103/physreve.91.012905] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/26/2014] [Indexed: 11/07/2022]
Abstract
A stability criterion for solitons of the driven nonlinear Schrödinger equation (NLSE) has been conjectured. The criterion states that p'(v)<0 is a sufficient condition for instability, while p'(v)>0 is a necessary condition for stability; here, v is the soliton velocity and p=P/N, where P and N are the soliton momentum and norm, respectively. To date, the curve p(v) was calculated approximately by a collective coordinate theory, and the criterion was confirmed by simulations. The goal of this paper is to calculate p(v) exactly for several classes and cases of the generalized NLSE: a soliton moving in a real potential, in particular a time-dependent ramp potential, and a time-dependent confining quadratic potential, where the nonlinearity in the NLSE also has a time-dependent coefficient. Moreover, we investigate a logarithmic and a cubic NLSE with a time-independent quadratic potential well. In the latter case, there is a bisoliton solution that consists of two solitons with asymmetric shapes, forming a bound state in which the shapes and the separation distance oscillate. Finally, we consider a cubic NLSE with parametric driving. In all cases, the p(v) curve is calculated either analytically or numerically, and the stability criterion is confirmed.
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Affiliation(s)
- Niurka R Quintero
- IMUS and Departamento de Física Aplicada I, E.S.P., Universidad de Sevilla, Virgen de África 7, E-41011 Sevilla, Spain
| | - Franz G Mertens
- Physikalisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
| | - A R Bishop
- Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Al Khawaja U, Boudjemâa A. Binding energy of soliton molecules in time-dependent harmonic potential and nonlinear interaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:036606. [PMID: 23031044 DOI: 10.1103/physreve.86.036606] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2012] [Indexed: 06/01/2023]
Abstract
We calculate the binding energy of soliton molecules of an integrable nonlinear Schro[over ̈]dinger equation with time-dependent harmonic potential and cubic nonlinearity. Through a scaling transformation, an exact formula for the binding energy can be derived from that of the free soliton molecules in a homogeneous background. In the special case of oscillatory time dependence, sharp resonances occur at some integer and fractional multiples of the natural frequency of the molecule. Enhanced binding is obtained at these resonances and over some finite continuous range of low frequencies.
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Affiliation(s)
- U Al Khawaja
- Physics Department, United Arab Emirates University, P.O. Box 17551, Al-Ain, United Arab Emirates
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Yan Z, Jiang D. Matter-wave solutions in Bose-Einstein condensates with harmonic and Gaussian potentials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:056608. [PMID: 23004896 DOI: 10.1103/physreve.85.056608] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/13/2011] [Revised: 02/13/2012] [Indexed: 06/01/2023]
Abstract
We study exact matter-wave solutions of the quasi-one-dimensional Gross-Pitaevskii (GP) equation with the space- and/or time-modulated potential and nonlinearity and the time-dependent gain or loss term in Bose-Einstein condensates. In particular, based on the similarity transformation and symbolic analysis, we report several families of exact solutions of the quasi-one-dimensional GP equation in the combination of the harmonic and Gaussian potentials, in which some physically relevant solutions are described. The stability of the obtained matter-wave solutions is addressed numerically such that some stable solutions are found. Moreover, we also analyze the parameter regimes for the stable solutions. These results may raise the possibility of relative experiments and potential applications.
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Affiliation(s)
- Zhenya Yan
- Key Laboratory of Mathematics Mechanization, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China.
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Sun ZY, Gao YT, Liu Y, Yu X. Soliton management for a variable-coefficient modified Korteweg-de Vries equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:026606. [PMID: 21929127 DOI: 10.1103/physreve.84.026606] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/08/2010] [Revised: 06/05/2011] [Indexed: 05/31/2023]
Abstract
The concept of soliton management has been explored in the Bose-Einstein condensate and optical fibers. In this paper, our purpose is to investigate whether a similar concept exists for a variable-coefficient modified Korteweg-de Vries equation, which arises in the interfacial waves in two-layer liquid and Alfvén waves in a collisionless plasma. Through the Painlevé test, a generalized integrable form of such an equation has been constructed under the Painlevé constraints of the variable coefficients based on the symbolic computation. By virtue of the Ablowitz-Kaup-Newell-Segur system, a Lax pair with time-dependent nonisospectral flow of the integrable form has been established under the Lax constraints which appear to be more rigid than the Painlevé ones. Under such Lax constraints, multisoliton solutions for the completely integrable variable-coefficient modified Korteweg-de Vries equation have been derived via the Hirota bilinear method. Moreover, results show that the solitons and breathers with desired amplitude and width can be derived via the different choices of the variable coefficients.
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Affiliation(s)
- Zhi-Yuan Sun
- Ministry-of-Education Key Laboratory of Fluid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
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He JR, Li HM. Analytical solitary-wave solutions of the generalized nonautonomous cubic-quintic nonlinear Schrödinger equation with different external potentials. Phys Rev E 2011; 83:066607. [PMID: 21797507 DOI: 10.1103/physreve.83.066607] [Citation(s) in RCA: 49] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/14/2010] [Revised: 01/04/2011] [Indexed: 11/07/2022]
Abstract
A large family of analytical solitary wave solutions to the generalized nonautonomous cubic-quintic nonlinear Schrödinger equation with time- and space-dependent distributed coefficients and external potentials are obtained by using a similarity transformation technique. We use the cubic nonlinearity as an independent parameter function, where a simple procedure is established to obtain different classes of potentials and solutions. The solutions exist under certain conditions and impose constraints on the coefficients depicting dispersion, cubic and quintic nonlinearities, and gain (or loss). We investigate the space-quadratic potential, optical lattice potential, flying bird potential, and potential barrier (well). Some interesting periodic solitary wave solutions corresponding to these potentials are then studied. Also, properties of a few solutions and physical applications of interest to the field are discussed. Finally, the stability of the solitary wave solutions under slight disturbance of the constraint conditions and initial perturbation of white noise is discussed numerically; the results reveal that the solitary waves can propagate in a stable way under slight disturbance of the constraint conditions and the initial perturbation of white noise.
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Affiliation(s)
- Jun-Rong He
- Department of Physics, Zhejiang Normal University, Jinhua, Zhejiang 321004, People's Republic of China
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