AAM  >> Vol. 5 No. 2 (May 2016)

    Lie Group Reduction for a Kind of Space-Fractional Order Nonlinear SchrO¨dinger Equation

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周春红,化存才:云南师范大学数学学院,云南 昆明

空间分数阶非线性SchrO¨dinger方程李群约化群不变解行波解Space-Fractional Order Nonlinear SchrO¨dinger Equation Lie Group Reduction Group-Invariant Solutions Travelling Wave Solutions



This paper will apply the Lie group reduction method to a kind of space-fractional order nonlinear Schrödinger equation. New single parameter solutions and reduced equations of Lie symmetry are obtained for the equation. Moreover, by solving the reduced equation of Lie symmetry, some group-invariant solutions and travelling wave solutions are given for the space-fractional order nonlinear Schrödinger equation.

周春红, 化存才. 一类空间分数阶非线性SchrO¨dinger方程的李群约化[J]. 应用数学进展, 2016, 5(2): 310-319. http://dx.doi.org/10.12677/AAM.2016.52039


[1] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York.
[2] Podlubny, I. (2002) Geometric and Physical Interpretation of Fractional Intergration and Fractional Differentiation. Fractional Calculus and Applied Analysis, 5, 367-386.
[3] Blair, G.W.S. (1947) The Role of Psychophysics in Rheology. Journal of Colloid Science, 2, 21-27.
[4] Hartley, T.T., Lorenzo, C.F. and Qammer, H.K. (1995) Chaos in Fractional Order Chua’s System. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42, 485-492.
[5] Caputo, M. (1993) Free Modes Splitting and Alterations of Electrochemically Polarizable Media. Journal Rendiconti Lincei, 4, 89-98.
[6] Cole, K. (1993) Electric Conductance of Biological System. Colorado Springs HAR, New York, 107-116.
[7] Anastasio, T.J. (1994) The Fractional-Order Dynamics of Brainstem Vestibulo-Oculomot or Neurons. Biological Cybernetics, 72, 69-76.
[8] 董建平. 分数阶微积分及其在分数阶量子力学中的应用[D]: [博士学位论文]. 济南: 山东大学, 2009.
[9] Muslih, S.I., Agrawal, O.P. and Baleanu, D. (2010) A Fractional Schrődinger Equation and Its Solution. International Journal of Theoretical Physics, 49, 1746-1752.
[10] Atangana, A. and Cloot, A.H. (2013) Stability and Convergence of the Space Fractional Variable-Order Schrődinger Equation. Advances in Difference Equations, 1, 80-90.
[11] Hu, Y. and Kallianpur, G. (2000) Schrődinger Equations with Fractional Laplacians. Applied Mathematics & Optimization, 42, 281-290.
[12] Liu, Y. (2015) Multiplicity of Solutions for Fractional Schrődinger Eq-uations with Perturbation. Boundary Value Problems, 1, 1-9.
[13] Dong, J.P. and Xu, M.Y. (2007) Solutions to the Space Fractional Schrődinger Equation Using Momentum Representation Method. Journal of Mathematical Physics, 48, 721-726.
[14] Secchi, S. (2013) Ground State Solutions for the Fractional Equation in RN. Journal of Mathematical Physics, 54, 315- 325.
[15] Zhang, J. and Zhu, S.H. (2015) Stability of Standing Waves for the Nonlinear Fractional Schrődinger Equation. Journal of Dynamics and Differential Equations, 477-501.
[16] 田畴. 李群及其在微分方程中的应用[M]. 北京: 科学出版社, 2005.
[17] 于兴江, 刘希强. 时间分数阶Boussinesq方程的李对称分析[J]. 物理学报, 2013, 62(23): 230201.
[18] Urban, R. and Kiewicz, J.Z. (2008) Fractional Time-Dependent Schrödinger Equation on the Heisenberg Group. Mathematische Zeitschrift, 260, 931-948.
[19] 杨绍杰. 时间分数阶KdV方程和耦合KdV方程组的Lie对称分析[D]: [硕士学位论文]. 昆明: 云南师范大学, 2013.
[20] Wu, G.C. (2010) A Fractional Lie Group Method for Anomalous Diffusion Equations. Communications on Fractional Calculus, 1, 27-31.
[21] 阮航宇, 李慧军. 用推广的李群约化法求解非线性薛定谔方程[J]. 物理学报, 2005, 54(3): 996-1001.
[22] Laskin, N. (2002) Fractional Schrödinger Equation. Physical Review E, 66, 561-569.
[23] Gaur, M. and Singh, K. (2014) On Group Invariant Solutions of Fractional Order Burgers-Poisson Equation. Applied Mathematics and Computation, 244, 870-877.
[24] Singh, J. and Kumar, D. (2014) New Analytical Approach for Frac-tional Cubic Nonlinear Schrödinger Equation via Laplace Transform. In: Babu, B.V., Nagar, A., Deep, K., Pant, M., Bansal, J.C., Ray, K. and Gupta, U., Eds., Proceedings of the Second International Conference on Soft Computing for Problem Solving (SocProS 2012), Springer, India, 271-277.
[25] Wu, G.C. (2011) Lie Group Classifications and Exact Solutions for Time-Fractional Burgers Equation. Communications in Theoretical Physics, 55, 1073-1081.