October  2011, 4(5): 1299-1325. doi: 10.3934/dcdss.2011.4.1299

Exact solutions for periodic and solitary matter waves in nonlinear lattices

1. 

Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong, China, China

2. 

Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel

Received  September 2009 Revised  November 2009 Published  December 2010

We produce three vast classes of exact periodic and solitonic solutions to the one-dimensional Gross-Pitaevskii equation (GPE) with the pseudopotential in the form of a nonlinear lattice (NL), induced by a spatially periodic modulation of the local nonlinearity. It is well known that NLs in Bose-Einstein condensates (BECs) may be created by means of the Feshbach-resonance technique. The model may also include linear potentials with the same periodicity. The NL modulation function, the linear potential (if any), and the corresponding exact solutions are expressed in terms of the Jacobi's elliptic functions of three types, cn, dn, and sn, which give rise to the three different classes of the solutions. The potentials and associated solutions are parameterized by two free constants and an additional sign parameter in the absence of the linear potential. In the presence of the latter, the solution families feature two additional free parameters. The families include both sign-constant and sign-changing NLs. Density maxima of the solutions may coincide with either minima or maxima of the periodic pseudopotential. The solutions reduce to solitons in the limit of the infinite period. The stability of the solutions is tested via systematic direct simulations of the GPE. As a result, stability regions are identified for the periodic solutions and solitons. The periodic patterns of cn type, and the respective limit-form solutions in the form of bright solitons, may be stable both in the absence and presence of the linear potential. On the contrary, the stability of the two other solution classes, of the dn and sn types, is only possible with the linear potential.
Citation: Cheng Hou Tsang, Boris A. Malomed, Kwok Wing Chow. Exact solutions for periodic and solitary matter waves in nonlinear lattices. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1299-1325. doi: 10.3934/dcdss.2011.4.1299
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show all references

References:
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F. K. Abdullaev and J. Garnier, Propagation of matter-wave solitons in periodic and random nonlinear potentials,, Phys. Rev. A, 72 (2005).  doi: 10.1103/PhysRevA.72.061605.  Google Scholar

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[9]

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[15]

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[16]

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[17]

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[18]

G. Chong and W. Hai, Dynamical evolutions of matter-wave bright solitons in an inverted parabolic potential,, J. Phys. B: At. Mol. Opt. Phys., 40 (2007), 211.  doi: 10.1088/0953-4075/40/1/019.  Google Scholar

[19]

K. W. Chow , C. K. Lam, K. Nakkeeran and B. Malomed, Transmission and stability of solitary pulses in complex Ginzburg - Landau equations with variable coefficients,, J. Phys. Soc. Jpn., 77 (2008).  doi: 10.1143/JPSJ.77.054001.  Google Scholar

[20]

K. W. Chow, B. A. Malomed and K. Nakkeeran, Exact solitary- and periodic-wave modes in coupled equations with saturable nonlinearity,, Phys. Lett. A, 359 (2006), 37.   Google Scholar

[21]

K. W. Chow, B. A. Malomed, B. Xiong and W. M. Liu, Singular nonlinearity management for matter-wave solitons in normal and inverted parabolic potential,, J. Phys. Soc. Jpn., 75 (2006).  doi: 10.1143/JPSJ.75.114004.  Google Scholar

[22]

K. W. Chow, K. Nakkeeran and B. A. Malomed, Periodic waves in bimodal optical fibers,, Opt. Commun., 219 (2003), 251.  doi: 10.1016/S0030-4018(03)01319-1.  Google Scholar

[23]

S. De Nicola, R. Fedele, D. Jovanovic, B. Malomed, M. A. Man'ko, V. I. Man'ko and P. K. Shukla, 1D Stability Analysis of Filtering and Controlling the Solitons in Bose-Einstein Condensates,, Eur. Phys. J. B, 54 (2006), 113.  doi: 10.1140/epjb/e2006-00418-0.  Google Scholar

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[25]

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[26]

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[27]

L. Fallani, C. Fort and M. Inguscio, Bose-Einstein condensates in optical potentials,, Rivista Nuovo Cim., 28 (2005), 1.   Google Scholar

[28]

G. Fibich, Y. Sivan and M. I. Weinstein, Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,, Physica D, 217 (2006), 31.  doi: 10.1016/j.physd.2006.03.009.  Google Scholar

[29]

J. Garnier and F. K. Abdullaev, Transmission of matter-wave solitons through nonlinear traps and barriers,, Phys. Rev. A, 74 (2006).  doi: 10.1103/PhysRevA.74.013604.  Google Scholar

[30]

W. Hai, Y. Li, B. Xia, and X. Luo, Exact solutions of a two-component BEC interacting with a lattice potential,, Europhys. Lett., 71 (2005), 28.  doi: 10.1209/epl/i2005-10070-x.  Google Scholar

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