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:
[1]

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.

[2]

U. Al. Khawaja, Integrability of a general Gross-Pitaevskii equation and exact solitonic solutions of a Bose-Einstein condensate in a periodic potential,, Phys. Lett. A, 373 (2009), 2710. doi: 10.1016/j.physleta.2009.05.049.

[3]

R. Atre, P. K. Panigrahi and G. S. Agarwal, Class of solitary wave solutions of the one-dimensional Gross-Pitaevskii equation,, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.056611.

[4]

J. Belmonte-Beitia, V. V. Konotop, V. M. Pérez-García, and V. E. Vekslerchik, Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: Linear and nonlinear lattices Chaos,, Solitons & Fractals, 41 (2009), 1158.

[5]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and V. V. Konotop, Localized nonlinear waves in systems with time- and space-modulated nonlinearities,, Phys. Rev. Lett., 100 (2008). doi: 10.1103/PhysRevLett.100.164102.

[6]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and P. J. Torres, Lie Symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,, Phys. Rev. Lett., 98 (2007). doi: 10.1103/PhysRevLett.98.064102.

[7]

J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik and P. J. Torres, Lie symmetries, qualitative analysis and exact solutions of nonlinear Schrödinger equations with inhomogeneous nonlinearities,, Discrete Contin. Dyn. Syst. B, 9 (2008), 221.

[8]

L. Bergé, V. K. Mezentsev, J. Juul Rasmussen, P. L. Christiansen and Yu. B. Gaididei, Self-guiding light in layered nonlinear media,, Opt. Lett., 25 (2000), 1037.

[9]

V. A. Brazhnyi and V. V. Konotop, Theory of nonlinear matter waves in optical lattices,, Mod. Phys. Lett. B, 18 (2004), 627. doi: 10.1142/S0217984904007190.

[10]

J. C. Bronski, L. D. Carr, R. Carretero-González, B. Deconinck, J. N. Kutz and K. Promislow, Stability of attractive Bose-Einstein condensates in a periodic potential,, Phys. Rev. E, 64 (2001).

[11]

J. C. Bronski, L. D. Carr, B. Deconinck and J. N. Kutz, Bose-Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential,, Phys. Rev. Lett., 86 (2001). doi: 10.1103/PhysRevLett.86.1402.

[12]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz and K. Promislow, Stability of repulsive Bose-Einstein condensates in a periodic potential,, Phys. Rev. E, 63 (2001). doi: 10.1103/PhysRevE.63.036612.

[13]

A. V. Carpentier, H. Michinel, M. I. Rodas-Verde and V. M. Pérez-García, Analysis of an atom laser based on the spatial control of the scattering length,, Phys. Rev. A, 74 (2006). doi: 10.1103/PhysRevA.74.013619.

[14]

L. D. Carr, C. W. Clark and W. P. Reinhardt, Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity,, Phys. Rev. A, 62 (2000). doi: 10.1103/PhysRevA.62.063610.

[15]

L. D. Carr, C. W. Clark and W. P. Reinhardt, Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity,, Phys. Rev. A, 62 (2000). doi: 10.1103/PhysRevA.62.063611.

[16]

R. Carretero-González, D. J. Frantzeskakis and P. G. Kevrekidis, Nonlinear waves in Bose-Einstein condensates: Physical relevance and mathematical techniques,, Nonlinearity, 21 (2008).

[17]

M. Centurion, M. A. Porter, P. G. Kevrekidis and D. Psaltis, Nonlinearity management in optics: Experiment, theory, and simulation,, Phys. Rev. Lett., 97 (2006). doi: 10.1103/PhysRevLett.97.033903.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

N. K. Efremidis and D. N. Christodoulides, Lattice solitons in Bose-Einstein condensates,, Phys. Rev. A, 67 (2003). doi: 10.1103/PhysRevA.67.063608.

[25]

G. A. El, A. Gammal and A. M. Kamchatnov, Oblique dark solitons in supersonic flow of a Bose-Einstein condensate,, Phys. Rev. Lett., 97 (2006). doi: 10.1103/PhysRevLett.97.180405.

[26]

C. J. Elliott and B. R. Suydam, Self-focusing phenomena in air-glass laser structures,, IEEE J. Quant. Electr., 11 (1975), 863. doi: 10.1109/JQE.1975.1068545.

[27]

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

[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.

[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.

[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.

[31]

W. A. Harrison, "Pseudopotentials in the Theory of Metals,", Benjamin, (1966). doi: 10.1209/epl/i2005-10070-x.

[32]

Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, Stable two-dimensional solitons in nonlinear lattices,, Opt. Lett., 34 (2009), 770. doi: 10.1364/OL.34.000770.

[33]

Y. V. Kartashov, V. A. Vysloukh, A. Szameit, F. Dreisow, Heinrich, S. Nolte, A. Tünnermann, T. Pertsch and L. Torner, Surface solitons at interfaces of arrays with spatially modulated nonlinearity,, Opt. Lett., 33 (2008), 1120. doi: 10.1364/OL.33.001120.

[34]

Y. V. Kartashov, V. A. Vysloukh and L. Torner, Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,, Opt. Lett., 33 (2008), 1747. doi: 10.1364/OL.33.001747.

[35]

Y. V. Kartashov, V. A. Vysloukh and L. Torner, Power-dependent shaping of vortex solitons in optical lattices with spatially modulated nonlinear refractive index,, Opt. Lett., 33 (2008), 2173. doi: 10.1364/OL.33.002173.

[36]

Y. S. Kivshar and G. P. Agrawal, "Optical Solitons,", Academic Press: San Diego, (2003).

[37]

L. Li, B. A. Malomed, D. Mihalache and W. M. Liu, Exact soliton-on-plane-wave solutions for two-component Bose-Einstein condensates,, Phys. Rev. E, 73 (2006). doi: 10.1103/PhysRevE.73.066610.

[38]

Z. X. Liang , Z. D. Zhang and W. M. Liu, Dynamics of a bright soliton in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential,, Phys. Rev. Lett., 94 (2005). doi: 10.1103/PhysRevLett.94.050402.

[39]

M. Machholm, C. J. Pethick and H. Smith, Band structure, elementary excitations, and stability of a Bose-Einstein condensate in a periodic potential,, Phys. Rev. A, 67 (2003). doi: 10.1103/PhysRevA.67.053613.

[40]

C. C. Mak, K. W. Chow and K. Nakkeeran, Soliton pulse propagation in averaged dispersion-managed optical fiber system,, J. Phys. Soc. Jpn., 74 (2005), 1449. doi: 10.1143/JPSJ.74.1449.

[41]

B. A. Malomed and M. Ya. Azbel, Modulational instability of a wave scattered by a nonlinear center,, Phys. Rev. B, 47 (1993), 10402. doi: 10.1103/PhysRevB.47.10402.

[42]

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