Exact Projective Excitations of Nonautonomous Nonlinear Schrödinger System in (1 + 1)-Dimensions
ABSTRACT
With the aid of a direct projective approach, a general transformation solution for the nonautonomous nonlinear Schr?dinger (NLS) system is derived. Based on certain known exact solutions of the projective equation, some periodic and localized excitations with novel properties are correspondingly revealed by entrancing appropriate system parameters. The integrable constraint conditions for the nonautonomous NLS system derived naturally here are consistent with the compatibility condition via the Painlevé analysis in other literature.
With the aid of a direct projective approach, a general transformation solution for the nonautonomous nonlinear Schr?dinger (NLS) system is derived. Based on certain known exact solutions of the projective equation, some periodic and localized excitations with novel properties are correspondingly revealed by entrancing appropriate system parameters. The integrable constraint conditions for the nonautonomous NLS system derived naturally here are consistent with the compatibility condition via the Painlevé analysis in other literature.
Cite this paper
J. Ye and C. Zheng, "Exact Projective Excitations of Nonautonomous Nonlinear Schrödinger System in (1 + 1)-Dimensions," Journal of Modern Physics, Vol. 3 No. 8, 2012, pp. 702-708. doi: 10.4236/jmp.2012.38095.
J. Ye and C. Zheng, "Exact Projective Excitations of Nonautonomous Nonlinear Schrödinger System in (1 + 1)-Dimensions," Journal of Modern Physics, Vol. 3 No. 8, 2012, pp. 702-708. doi: 10.4236/jmp.2012.38095.
References
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[1] K. E. Strecker, G. B. Partridge, A. G. Truscott and R. G. Hulet, “Formation and Propagation of Matter-Wave Soli- ton Trains,” Nature, Vol. 417, 2002, pp. 150-153. doi:10.1038/nature747 [2] L. F. Mollenauer, R. H. Stolen and J. P. Gordon, “Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers,” Physics Review Letter, Vol. 45, No. 13, 1980, pp. 1095-1098. doi:10.1103/PhysRevLett.45.1095 [3] C. Q. Dai, X. G. Wang and J. F. Zhang, “Nonautonomous Spatiotemporal Localized Structures in the Inhomogene- ous Optical Fibers: Interaction and Control,” Annals of Physics, Vol. 326, No. 3, 2011, pp. 645-656. doi:10.1016/j.aop.2010.11.005 [4] D. Zhao, X. G. He and H. G. Luo, “Transformation from the Nonautonomous to Standard NLS Equations,” Physics and Astronomy, Vol. 53, No. 2, 2009, pp. 213-216. doi:10.1140/epjd/e2009-00051-7 [5] I. Towers and B. A. Malomed, “Stable (2+1)-Dimen- sional Solitons in a Layered Medium with Sign-Alter- nating Kerr Nonlinearity,” Journal of the Optical Society of America B, Vol. 19, 2002, pp. 537-543. doi:10.1364/JOSAB.19.000537 [6] Y. Gao and S. Y. Lou, “Analytical Solitary Wave Solutions to a (3+1)-Dimensional Gross-Pitaevskii Equation with Variable Coefficients,” Communications in Theore- tical Physics, Vol. 52, 2009, pp. 1030-1035. [7] V. N. Serkin, A. Hasegawa and T. L. Belyaeva, “No- nautonomous Solitons in External Potentials,” Physics Review Letter, Vol. 98, No. 7, 2007, Article ID: 074102. doi:10.1103/PhysRevLett.98.074102 [8] S. A. Ponomarenko and G. P. Agrawal, “Do Solitonlike Self-Similar Waves Exist in Nonlinear Optical Media?” Physics Review Letter, Vol. 97, No. 1, 2006, Article ID: 013901. doi:10.1103/PhysRevLett.97.013901 [9] S. Y. Lou, H. C. Hu and X. Y. Tang, “Interactions among Periodic Waves and Solitary Waves of the (N+1)-Di- mensional Sine-Gordon Field,” Physics Review E, Vol. 71, No. 3, 2005, Article ID: 036604. doi:10.1103/PhysRevE.71.036604 [10] S. Y. Lou and G. J. Ni, “The Relations among a Special Type of Solutions in Some (D+1)-Dimensional Nonlinear Equations,” Journal of Mathematical Physics, Vol. 30, No. 7, 1989, pp. 1614-1620. doi:10.1063/1.528294 [11] H. M. Li, “Searching for (3+1)-Dimensional Painlevé Integrable Model and Its Solitary Wave Solution,” Chi- nese Physics Letters, Vol. 19, No. 6, 2002, pp. 745-747. doi:10.1088/0256-307X/19/6/301 [12] C. L. Zheng, H. P. Zhu and L. Q. Chen, “Exact Solution and Semifolded Structures of Generalized BroerKaup System in (2+1)-Dimensions,” Chaos, Solitons & Fractals, Vol. 26, No. 1, 2004, pp. 181-194. doi:10.1016/j.chaos.2004.12.017 [13] Sirendaoreji and S. Jiong, “Auxiliary Equation Method for Solving Nonlinear Partial Differential Equations,” Physics Letters A, Vol. 309, No. 5-6, 2003, pp. 387-396. doi:10.1016/S0375-9601(03)00196-8 [14] E. G. Fan, “An Algebraic Method for Finding a Series of Exact Solutions to Integrable and Nonintegrable Nonlinear Evolution Equations,” Journal of Physics A: Mathematical and General, Vol. 36, No. 25, 2003, p. 7009. doi:10.1088/0305-4470/36/25/308 [15] C. L. Zheng and L. Q. Chen, “Some Novel Evolutional Behaviors of Localized Excitations in the Boiti-Leon- Martina-Pempinelli System,” International Journal of Modern Physics B, Vol. 22, No. 6, 2008, pp. 671-682. doi:10.1142/S0217979208038879 [16] H. Y. Wu, J. X. Fei and C. L. Zheng, “Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schr?dinger Equation with an External Potential,” Communications in Theoretical Physics, Vol. 54, 2010, pp. 55-59. [17] J. X. Fei and C. L. Zheng, “Chirped Self-Similar Solutions of a Generalized Nonlinear Schr?dinger Equation,” Verlag der Zeitschrift für Naturforschung, Vol. 66, 2011, pp. 1-5. [18] C. Q. Dai, Y. Y. Wang and X. G. Wang, “Ultrashort Self-Similar Solutions of the Cubic-Quintic Nonlinear Schr?dinger Equation with Distributed Coefficients in the Inhomogeneous Fiber,” Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 15, 2011, Article ID: 155203. doi:10.1088/1751-8113/44/15/155203 [19] C. Q. Dai, R. P. Chen and G. Q. Zhou, “Spatial Solitons with the Odd and Even Symmetries in (2+1)-Dimensional Spatially Inhomogeneous Cubic-Quintic Nonlinear Media with Transverse W-Shaped Modulation,” Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 44, No. 14, 2011, Article ID: 145401. doi:10.1088/0953-4075/44/14/145401 [20] C. Sulem and P. L. Sulem, “The Nonlinear Schr?dinger Equation: Self-focusing and Wave Collapse,” Springer- Verlag, New York, 1991. [21] F. Calogero, A. Degasperis and J. Xiaoda, “Nonlinear Schr?dinger-Type Equations from Multiscale Reduction of PDEs. I. Systematic Derivation,” Journal of Mathematical Physics, Vol. 41, No. 9, 2000, p. 6399. doi:10.1063/1.1287644 [22] F. Calogero and A. Degasperis, “Nonlinear Schr?dinger- type Equations from Multiscale Reduction of PDEs. II. Necessary Conditions of Integrability for Real PDEs,” Journal of Mathematical Physics, Vol. 42, No. 6, 2001, pp. 2635-2652. doi:10.1063/1.1366296 [23] J. He and Y. Li, “Designable Integrability of the Variable Coefficient Nonlinear Schr?dinger Equations,” Studies in Applied Mathematics, Vol. 126, No. 1, 2011, pp. 1-15. doi:10.1111/j.1467-9590.2010.00495.x [24] J. B. Beitia, V. M. Perez-Garcia and V. Brazhnyib, “Solitary Waves in Coupled Nonlinear Schr?dinger Equations with Spatially Inhomogeneous Nonlinearities,” Commu- nications in Nonlinear Science and Numerical Simulation, Vol. 16, No. 1, 2011, Article ID: 158172. doi:10.1016/j.cnsns.2010.02.024 [25] L. Gagnon and P. Winternitz, “Symmetry Classes of Variable Coefficient Nonlinear Schr?dinger Equations,” Journal of Physics A: Mathematical and General, Vol. 26, No. 23, 1993, p. 7061. doi:10.1088/0305-4470/26/23/043