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Transition to chaos in discrete nonlinear Schrödinger equation with long-range interaction

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  • Korabel, Nickolay
  • Zaslavsky, George M.

Abstract

Discrete nonlinear Schrödinger (DNLS) equation describes a chain of oscillators with nearest-neighbor interactions and a specific nonlinear term. We consider its modification with long-range interaction through a potential proportional to 1/l1+α with fractional α<2 and l as a distance between oscillators. This model is called αDNLS. It exhibits competition between the nonlinearity and a level of correlation between interacting far-distanced oscillators, that is defined by the value of α. We consider transition to chaos in this system as a function of α and nonlinearity. It is shown that decreasing of α with respect to nonlinearity stabilize the system. Connection of the model to the fractional generalization of the NLS (called FNLS) in the long-wave approximation is also discussed and some of the results obtained for αDNLS can be correspondingly extended to the FNLS.

Suggested Citation

  • Korabel, Nickolay & Zaslavsky, George M., 2007. "Transition to chaos in discrete nonlinear Schrödinger equation with long-range interaction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(2), pages 223-237.
    • Handle: RePEc:eee:phsmap:v:378:y:2007:i:2:p:223-237
    DOI: 10.1016/j.physa.2006.10.041
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    References listed on IDEAS

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    1. Cai, David & McLaughlin, David W. & Shatah, Jalal, 2001. "Spatiotemporal chaos in spatially extended systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 55(4), pages 329-340.
    2. Herbst, B.M. & Varadi, F. & Ablowitz, M.J., 1994. "Symplectic methods for the nonlinear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 37(4), pages 353-369.
    3. Goldfain, Ervin, 2006. "Complexity in quantum field theory and physics beyond the standard model," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 913-922.
    4. Ablowitz, Mark J. & Schober, Constance M., 1994. "Homoclinic manifolds and numerical chaos in the nonlinear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 37(4), pages 249-264.
    5. Tarasov, Vasily E. & Zaslavsky, George M., 2005. "Fractional Ginzburg–Landau equation for fractal media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 354(C), pages 249-261.
    6. Ablowitz, M.J. & Herbst, B.M. & Schober, C.M., 1996. "Computational chaos in the nonlinear Schrödinger equation without homoclinic crossings," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 228(1), pages 212-235.
    7. Lima, Rodrigo P.A & Lyra, Marcelo L & Cressoni, José C, 2001. "Multifractality of one electron eigen states in 1D disordered long range models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 295(1), pages 154-157.
    8. Ablowitz, M.J. & Herbst, B.M. & Schober, C.M., 1997. "The nonlinear Schrödinger equation: Asymmetric perturbations, traveling waves and chaotic structures," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 43(1), pages 3-12.
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    1. Tarasov, Vasily E. & Zaslavsky, George M., 2007. "Fractional dynamics of systems with long-range space interaction and temporal memory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 383(2), pages 291-308.

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