Exponential time differencing schemes for the 3-coupled nonlinear fractional Schrödinger equation
Tóm tắt
Two modified exponential time differencing schemes based on the Fourier spectral method are developed to solve the 3-coupled nonlinear fractional Schrödinger equation. We compare the stability of the schemes by plotting their stability regions. The local truncation errors of the time integrators are proved to be fourth-order. Numerical experiments illustrating the solution to the equations with various parameters and the mass conservation results of the numerical methods are carried out.
Tài liệu tham khảo
Longhi, S.: Fractional Schrödinger equation in optics. Opt. Lett. 40, 1117–1120 (2015)
Antoine, X., Tang, Q., Zhang, Y.: On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross–Pitaevskii equations with rotation term and nonlocal nonlinear interactions. J. Comput. Phys. 325, 74–97 (2016)
Fu, C., Lu, C.N., Yang, H.W.: Time-space fractional \((2 + 1)\) dimensional nonlinear Schrödinger equation for envelope gravity waves in baroclinic atmosphere and conservation laws as well as exact solutions. Adv. Differ. Equ. 2018, 56 (2018)
Petroni, N.C., Pusterla, M.: Lévy processes and Schrödinger equation. Physica A 388, 824–836 (2009)
Wang, P., Huang, C.: A conservative linearized difference scheme for the nonlinear fractional Schrödinger equations. Numer. Algorithms 69, 625–641 (2015)
Wang, P., Huang, C.: An energy conservative difference scheme for the nonlinear fractional Schrödinger equations. J. Comput. Phys. 293, 238–251 (2015)
Wang, P., Huang, C.: Split-step alternating direction implicit difference scheme for the fractional Schrödinger equation in two dimensions. Comput. Math. Appl. 71, 1114–1128 (2016)
Duo, S., Zhang, Y.: Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation. Comput. Math. Appl. 71, 2257–2271 (2016)
Dehghan, M., Abbaszadeh, M., Mohebbi, A.: Numerical solution of system of n-coupled nonlinear Schrödinger equations via two variants of the meshless local Petrov–Galerkin (MLPG) method. Comput. Model. Eng. Sci. 100, 399–444 (2014)
Taleei, A., Dehghan, M.: Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations. Comput. Phys. Commun. 185, 1515–1528 (2014)
Mohebbi, A., Abbaszadeh, M., Dehghan, M.: The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics. Eng. Anal. Bound. Elem. 37, 475–485 (2013)
Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. 26, 448–479 (2010)
Dehghan, M.: Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices. Math. Comput. Simul. 71, 16–30 (2006)
Dehghan, M., Fakhar-Izadi, F.: The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves. Math. Comput. Model. 53, 1865–1877 (2011)
Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34, 200–218 (2010)
Wang, D., Xiao, A., Yang, W.: A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations. J. Comput. Phys. 272, 644–655 (2014)
Weng, Z.F., Zhai, S.Y., Feng, X.L.: A Fourier spectral method for fractional-in-space Cahn–Hilliard equation. Appl. Math. Model. 42, 462–477 (2017)
Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)
Bhatt, H.P., Khaliq, A.Q.M.: Fourth-order compact schemes for the numerical simulation of coupled Burgers’ equation. Comput. Phys. Commun. 200, 117–138 (2016)
Bueno-Orovio, A., Kay, D.: Fourier spectral methods for fractional-in-space reaction–diffusion equations. BIT Numer. Math. 54, 937–954 (2014)
Yousuf, M., Khaliq, A.Q.M., Kleefeld, B.: The numerical approximation of nonlinear Black–Scholes model for exotic path-dependent American options with transaction cost. Int. J. Comput. Math. 89, 1239–1254 (2012)
Liang, X., Khaliq, A.Q.M., Bhatt, H., Furati, K.M.: The locally extrapolated exponential splitting scheme for multi-dimensional nonlinear space-fractional Schrödinger equations. Numer. Algorithms 76, 939–958 (2017)
Beylkin, G., Keiser, J.M., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147, 362–387 (1998)
Hederi, M., Islas, A.L., Reger, K., Schober, C.M.: Efficiency of exponential time differencing schemes for nonlinear Schrödinger equations. Math. Comput. Simul. 127, 101–113 (2016)
Aydin, A.: Multisymplectic integration of n-coupled nonlinear Schrödinger equation with destabilized periodic wave solutions. Chaos Solitons Fractals 41, 735–751 (2009)
Qian, X., Song, S., Chen, Y.: A semi-explicit multi-symplectic splitting scheme for a 3-coupled nonlinear Schrödinger equation. Comput. Phys. Commun. 185, 1255–1264 (2014)