Conservative Fourier spectral method and numerical investigation of space fractional Klein-Gordon-Schrödinger equations

Abstract In this paper, we propose Fourier spectral method to solve space fractional Klein–Gordon–Schrodinger equations with periodic boundary condition. First, the semi-discrete scheme is given by using Fourier spectral method in spatial direction, and conservativeness and convergence of the semi-discrete scheme are discussed. Second, the fully discrete scheme is obtained based on Crank–Nicolson/leap-frog methods in time direction. It is shown that the scheme can be decoupled, and preserves mass and energy conservation laws. It is proven that the scheme is of the accuracy O ( τ 2 + N − r ) . Last, based on the numerical experiments, the correctness of theoretical results is verified, and the effects of the fractional orders α, β on the solitary solution behaviors are investigated. In particular, some interesting phenomena including the quantum subdiffusion are observed, and complex dynamical behaviors are shown clearly by many intuitionistic images.

[1]  Luming Zhang,et al.  A class of conservative orthogonal spline collocation schemes for solving coupled Klein-Gordon-Schrödinger equations , 2008, Appl. Math. Comput..

[2]  Wei Zeng,et al.  Finite Difference/Finite Element Methods for Distributed-Order Time Fractional Diffusion Equations , 2017, J. Sci. Comput..

[3]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[4]  Fanhai Zeng,et al.  Numerical Methods for Fractional Calculus , 2015 .

[5]  Chengming Huang,et al.  Point-wise error estimate of a conservative difference scheme for the fractional Schrödinger equation , 2016, J. Comput. Appl. Math..

[6]  Wei Yang,et al.  Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative , 2013, J. Comput. Phys..

[7]  Bazar Babajanov,et al.  The Fractional Schrodinger-Klein-Gordon Equation and Intermediate Relativism , 2013 .

[8]  C. Pozrikidis.  The Fractional Laplacian , 2016 .

[9]  MOHSEN ZAYERNOURI,et al.  Spectral and Discontinuous Spectral Element Methods for Fractional Delay Equations , 2014, SIAM J. Sci. Comput..

[10]  Yangquan Chen,et al.  High-order algorithms for Riesz derivative and their applications (II) , 2015, J. Comput. Phys..

[11]  K. Burrage,et al.  Fourier spectral methods for fractional-in-space reaction-diffusion equations , 2014 .

[12]  S. C. Lim,et al.  Finite temperature Casimir effect for a massless fractional Klein-Gordon field with fractional Neumann conditions , 2007, 0804.3915.

[13]  Yali Duan,et al.  Semi-explicit symplectic partitioned Runge-Kutta Fourier pseudo-spectral scheme for Klein-Gordon-Schrödinger equations , 2010, Comput. Phys. Commun..

[14]  Luming Zhang.  Convergence of a conservative difference scheme for a class of Klein-Gordon-Schrödinger equations in one space dimension , 2005, Appl. Math. Comput..

[15]  N. Laskin.  Fractional quantum mechanics and Lévy path integrals , 1999, hep-ph/9910419.

[16]  Chengming Huang,et al.  An energy conservative difference scheme for the nonlinear fractional Schrödinger equations , 2015, J. Comput. Phys..

[17]  Aiguo Xiao,et al.  Fourier pseudospectral method on generalized sparse grids for the space-fractional Schrödinger equation , 2018, Comput. Math. Appl..

[18]  J. P. Roop.  Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R 2 , 2006 .

[19]  Aiguo Xiao,et al.  Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients , 2010, J. Comput. Appl. Math..

[20]  V. E. Tarasov.  Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media , 2011 .

[21]  Luming Zhang,et al.  High-order linear compact conservative method for the nonlinear Schrödinger equation coupled with the nonlinear Klein–Gordon equation , 2013 .

[22]  Weizhu Bao,et al.  A uniformly accurate (UA) multiscale time integrator Fourier pseudospectral method for the Klein–Gordon–Schrödinger equations in the nonrelativistic limit regime , 2015, Numerische Mathematik.

[23]  Mingliang Wang,et al.  The periodic wave solutions for the Klein–Gordon–Schrödinger equations , 2003 .

[24]  Jie Shen,et al.  Generalized Jacobi functions and their applications to fractional differential equations , 2014, Math. Comput..

[25]  Ai-guo Xiao,et al.  An efficient conservative difference scheme for fractional Klein-Gordon-Schrödinger equations , 2018, Appl. Math. Comput..

[26]  Edson Pindza,et al.  Fourier spectral method for higher order space fractional reaction-diffusion equations , 2016, Commun. Nonlinear Sci. Numer. Simul..

[27]  Cécile Piret,et al.  A Chebyshev PseudoSpectral Method to Solve the Space-Time Tempered Fractional Diffusion Equation , 2014, SIAM J. Sci. Comput..

[28]  N. Laskin,et al.  Fractional quantum mechanics , 2008, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  Hua Liang,et al.  Linearly implicit conservative schemes for long-term numerical simulation of Klein-Gordon-Schrödinger equations , 2014, Appl. Math. Comput..

[30]  Jie Shen,et al.  Spectral Methods: Algorithms, Analysis and Applications , 2011 .

[31]  Wei Yang,et al.  A linearly implicit conservative difference scheme for the space fractional coupled nonlinear Schrödinger equations , 2014, J. Comput. Phys..

[32]  Yanzhi Zhang,et al.  Mass-conservative Fourier spectral methods for solving the fractional nonlinear Schrödinger equation , 2016, Comput. Math. Appl..

[33]  G. Bo-ling,et al.  Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data , 2016 .

[34]  Abdul-Qayyum M. Khaliq,et al.  An efficient Fourier spectral exponential time differencing method for the space-fractional nonlinear Schrödinger equations , 2018, Comput. Math. Appl..

[35]  Mohamed M. Khader,et al.  An efficient approximate method for solving linear fractional Klein–Gordon equation based on the generalized Laguerre polynomials , 2013, Int. J. Comput. Math..

[36]  X. Liang,et al.  A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations , 2017, Numerical Algorithms.

[37]  Fawang Liu,et al.  A Crank-Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation , 2014, SIAM J. Numer. Anal..

[38]  Cem Çelik,et al.  Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative , 2012, J. Comput. Phys..

[39]  Xueke Pu,et al.  Fractional Partial Differential Equations and their Numerical Solutions , 2015 .

[40]  A. Pastor,et al.  Orbital stability of periodic waves for theKlein-Gordon-Schrödinger system , 2011 .

[41]  F. Polito,et al.  Fractional Klein–Gordon Equations and Related Stochastic Processes , 2013, 1308.2139.

[42]  A. Pastor,et al.  Stability Properties of Periodic Standing Waves for the Klein-Gordon-Schrodinger System , 2009, 0907.2142.

[43]  R. Ansari,et al.  Bright and singular soliton solutions of the conformable time-fractional Klein–Gordon equations with different nonlinearities , 2018 .

[44]  Xuan Zhao,et al.  A Fourth-order Compact ADI scheme for Two-Dimensional Nonlinear Space Fractional Schrödinger Equation , 2014, SIAM J. Sci. Comput..

[45]  V. Uchaikin.  Fractional Derivatives for Physicists and Engineers , 2013 .

[46]  Tingchun Wang,et al.  Point-wise errors of two conservative difference schemes for the Klein–Gordon–Schrödinger equation , 2012 .

[47]  Weihua Deng,et al.  High order schemes for the tempered fractional diffusion equations , 2016, Adv. Comput. Math..

[48]  K. Boubaker,et al.  An attempt to give exact solitary and periodic wave polynomial solutions to the nonlinear Klein–Gordon–Schrödinger equations , 2015 .

[49]  Chengjian Zhang,et al.  A conservative difference scheme for solving the strongly coupled nonlinear fractional Schrödinger equations , 2016, Commun. Nonlinear Sci. Numer. Simul..

[50]  Chengming Huang,et al.  Galerkin finite element method for the nonlinear fractional Ginzburg–Landau equation , 2017 .

[51]  Sunil Kumar,et al.  An accurate numerical method for solving the linear fractional Klein–Gordon equation , 2014 .

[52]  Dumitru Baleanu,et al.  A Spectral Legendre–Gauss–Lobatto Collocation Method for a Space-Fractional Advection Diffusion Equations with Variable Coefficients , 2013 .

[53]  Y. Yang.  Jacobi spectral Galerkin methods for fractional integro-differential equations , 2015 .

[54]  Wei Yang,et al.  Maximum-norm error analysis of a difference scheme for the space fractional CNLS , 2015, Appl. Math. Comput..

[55]  Li Yang,et al.  Efficient and accurate numerical methods for the Klein-Gordon-Schrödinger equations , 2007, J. Comput. Phys..

[56]  Dumitru Baleanu,et al.  On nonlinear fractional Klein-Gordon equation , 2011, Signal Process..

[57]  Yuepeng Wang,et al.  Generalized solitary wave solutions for the Klein-Gordon-Schrödinger equations , 2009, Comput. Math. Appl..