空间分数阶Schr?dinger方程调制不稳定性的数值研究
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摘要
调制不稳定性在数学和物理等学科中应用十分广泛.本文主要通过分裂谱方法对空间分数阶薛定谔方程进行数值计算,并根据Benjamin-FeirLighthill准则推导了非线性薛定谔方程的调制不稳定条件.文中分别研究了空间分数阶薛定谔方程在不同初值条件下的不稳定行为,并与整数阶薛定谔方程的不稳定性行为作比较,通过数值比较分析,发现整数阶薛定谔方程的这种不稳定行为对于空间分数阶薛定谔方程同样存在.
Modulational instability is widely used in mathematics and physics. In this work, we mainly use splitting Fourier spectral method to numerical calculate the space fractional Schr?dinger equation,we deduced the modulational instability condition of space fractional Schr?dinger equation by the Benjamin-Feir-Lighthill criterion. And we studied the different modulational instability behavior of space fractional Schr?dinger equation in different initial conditions, we also compare to the integer order Schr?dinger equation. Compared with the modulational instability behavior of integer order,it turns out that this modulational instability behavior applies to fractional Schr?dinger equations as well.
Modulational instability is widely used in mathematics and physics. In this work, we mainly use splitting Fourier spectral method to numerical calculate the space fractional Schr?dinger equation,we deduced the modulational instability condition of space fractional Schr?dinger equation by the Benjamin-Feir-Lighthill criterion. And we studied the different modulational instability behavior of space fractional Schr?dinger equation in different initial conditions, we also compare to the integer order Schr?dinger equation. Compared with the modulational instability behavior of integer order,it turns out that this modulational instability behavior applies to fractional Schr?dinger equations as well.
引文
[1] Zakharov V E, Ostrovsky L A. Modulation instability:the beginning[J]. Phys. D, 2009,238:540-548.
[2] Arshad M, Seadawy Aly R, Lu Dianchen. Exact brightdark solitary wave solutions of the higherorder cubicquintic nonlinear Schr¨odinger equation and its stability[J]. Optik, 2017,138:40-49.
[3] Biondini G, Li S, Mantzavinos D, et al. Universal behavior of modulationally unstable media[J].SIAM Rev., 2018,60(4):888-908.
[4] Zhang L, He Z, Conti C, et al. Modulational instability in fractional nonlinear Schroedinger equation[J]. Communications in Nonlinear, Science and Numerical Simulation, 2017,48:531-540.
[5] Kwas′nicki M. Ten equivalent definitions of the fractional Laplace operator[J]. Fractional Calculus and Applied Analysis, 2017,20(1):7-51.
[6] Gatto P, Hesthaven J S. Numerical approximation of the fractional Laplacian via hp-finite elements,with an application to image denoising[J]. J. Sci. Comput., 2015(65):249-270.
[7] Feynman R P, Hibbs A R. Quantum Mechanics and Path Integrals[M]. New York:McGraw-Hill,1965.
[8] Laskin N. Fractional Schr¨odinger equation[J]. Phys. Rev. E., 2002,66:249-264.
[9] Guo X, Xu M. Some physical applications of fractional Schr¨odinger equation[J]. J. Math. Phys.,2006,47:1-9.
[10] Kirkpatrick K, Lenzmann E, Staffilani G. On the continuum limit for discrete NLS with long-range lattice interactions[J]. Comm. Math. Phys., 2013,317:563-591.
[11] Longhi S. Fractional Schr¨odinger equation in optics[J]. Opt. Lett., 2015,40:1117-1120.
[12] Strang G. On the construction and comparison of difference schemes[J]. SIAM J. Numer. Anal.,1968,5:506-517.
[13] Zhai S Y, Weng Z F, Feng X L. Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen-Cahn model[J]. Appl. Math. Model., 2016,40(2):1315-1324.
[14] Cheng Y Z, Kurganov A, Qu Z L, et al. Fast and stable explicit operator splitting methods for phase-field models[J]. J. Comput. Phys., 2015,303:45-65.
[2] Arshad M, Seadawy Aly R, Lu Dianchen. Exact brightdark solitary wave solutions of the higherorder cubicquintic nonlinear Schr¨odinger equation and its stability[J]. Optik, 2017,138:40-49.
[3] Biondini G, Li S, Mantzavinos D, et al. Universal behavior of modulationally unstable media[J].SIAM Rev., 2018,60(4):888-908.
[4] Zhang L, He Z, Conti C, et al. Modulational instability in fractional nonlinear Schroedinger equation[J]. Communications in Nonlinear, Science and Numerical Simulation, 2017,48:531-540.
[5] Kwas′nicki M. Ten equivalent definitions of the fractional Laplace operator[J]. Fractional Calculus and Applied Analysis, 2017,20(1):7-51.
[6] Gatto P, Hesthaven J S. Numerical approximation of the fractional Laplacian via hp-finite elements,with an application to image denoising[J]. J. Sci. Comput., 2015(65):249-270.
[7] Feynman R P, Hibbs A R. Quantum Mechanics and Path Integrals[M]. New York:McGraw-Hill,1965.
[8] Laskin N. Fractional Schr¨odinger equation[J]. Phys. Rev. E., 2002,66:249-264.
[9] Guo X, Xu M. Some physical applications of fractional Schr¨odinger equation[J]. J. Math. Phys.,2006,47:1-9.
[10] Kirkpatrick K, Lenzmann E, Staffilani G. On the continuum limit for discrete NLS with long-range lattice interactions[J]. Comm. Math. Phys., 2013,317:563-591.
[11] Longhi S. Fractional Schr¨odinger equation in optics[J]. Opt. Lett., 2015,40:1117-1120.
[12] Strang G. On the construction and comparison of difference schemes[J]. SIAM J. Numer. Anal.,1968,5:506-517.
[13] Zhai S Y, Weng Z F, Feng X L. Fast explicit operator splitting method and time-step adaptivity for fractional non-local Allen-Cahn model[J]. Appl. Math. Model., 2016,40(2):1315-1324.
[14] Cheng Y Z, Kurganov A, Qu Z L, et al. Fast and stable explicit operator splitting methods for phase-field models[J]. J. Comput. Phys., 2015,303:45-65.