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Original Articles

Analysis of a Fourier pseudo-spectral conservative scheme for the Klein–Gordon–Schrödinger equation

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Pages 36-60 | Received 02 Jan 2017, Accepted 28 Jun 2017, Published online: 23 Aug 2017

References

  • W. Bao and L. Yang, Efficient and accurate numerical methods for the Klein–Gordon–Schrödinger equations, J. Comput. Phys. 225 (2007), pp. 1863–1893. doi: 10.1016/j.jcp.2007.02.018
  • T. Bridges and S. Reich, Multi-symplectic spectral discretizations for the Zakharov–Kuznetsov and shallow water equations, Phys. D 152–153 (2001), pp. 491–504. doi: 10.1016/S0167-2789(01)00188-9
  • T. Bridges and S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A 284 (2001), pp. 184–193. doi: 10.1016/S0375-9601(01)00294-8
  • J. Cai, J. Wang, and Y. Wang, A local energy-preserving scheme for Klein--Gordon--Schr equations, Chin. Phys. B 24 (2015), p. 050205.
  • C. Canuto, M. Hussaini, A. Quarteroni, M. Hussaini, and T. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006.
  • C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comput. 38 (1982), pp. 67–86. doi: 10.1090/S0025-5718-1982-0637287-3
  • J. Chen and M. Qin, Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation, Electr. Trans. Numer. Anal. 12 (2001), pp. 193–204.
  • A. Darwish and E. Fan, A series of new explicit exact solutions for the coupled Klein–Gordon–Schrödinger equations, Chaos Solitons Fractals 20 (2004), pp. 609–617. doi: 10.1016/S0960-0779(03)00419-3
  • M. Dehghan, M. Abbaszadeh, and A. Mohebbi, 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 (2014), pp. 399–444.
  • M. Dehghan and V. Mohammadi, Two numerical meshless techniques based on radial basis functions (RBFs) and the method of generalized moving least squares (GMLS) for simulation of coupled Klein–Gordon–Schrödinger (KGS) equations, Comput. Math. Appl. 71 (2016), pp. 892–921. doi: 10.1016/j.camwa.2015.12.033
  • M. Dehghan and A. Taleei, Numerical solution of the Yukawa-coupled Klein–Gordon–Schrödinger equations via a Chebyshev pseudospectral multidomain method, Appl. Math. Model. 36 (2012), pp. 2340–2349. doi: 10.1016/j.apm.2011.08.030
  • K. Feng and M. Qin, Symplectic Geometric Algorithms for Hamiltonian Systems, Springer-Verlag/Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, Berlin/Hangzhou, 2003.
  • I. Fukuda and M. Tsutsumi, On coupled Klein–Gordon–Schroödinger equations II, J. Math. Anal. Appl. 66 (1978), pp. 358–378. doi: 10.1016/0022-247X(78)90239-1
  • Y. Gong, J. Cai, and Y. Wang, Multi-symplectic Fourier pseudospectral method for the Kawahara equation, Commun. Comput. Phys. 16 (2014), pp. 35–55. doi: 10.4208/cicp.090313.041113a
  • Y. Gong, Q. Wang, Y. Wang, and J. Cai, A conservative Fourier pseudo-spectral method for the nonlinear Schrödinger equation, J. Comput. Phys. 328 (2017), pp. 354–370. doi: 10.1016/j.jcp.2016.10.022
  • O. Gonzalez, Time integration and discrete Hamiltonian systems, J. Nonlinear Sci. 6 (1996), pp. 449–467. doi: 10.1007/BF02440162
  • B. Guo, Global solution for some problem of a class of equations in interaction of complex Schrödinger field and real Klein–Gordon field, Sci. China Ser. A 2 (1982), pp. 97–107.
  • W. Guo, Z. Zhang, and H. Ma, Optimal error estimates of multi-symplectic method for nonlinear Schrödinger equation, J. Shanghai Univ. 15 (2009), pp. 487–492.
  • P. Hansen, J. Nagy, and D. OLeary, Deblurring Images: Matrices, Spectra and Filtering, Chapter 4, SIAM, Philadelphia, 2006.
  • N. Hayashi and W. Wahl, On the global strong solutions of coupled Klein–Gordon–Schrödinger equations, J. Math. Soc. Japan 39 (1987), pp. 489–497. doi: 10.2969/jmsj/03930489
  • F. Hioe, Periodic solitary waves for two coupled nonlinear Klein–Gordon–Schrödinger equations, J. Phys. A 36 (2003), pp. 7307–7330. doi: 10.1088/0305-4470/36/26/307
  • J. Hong, S. Jiang, L. Kong, and C. Li, Numerical comparison of five difference schemes for coupled Klein–Gordon–Schrödinger equations in quantum physics, J. Phys. A 40 (2007), pp. 9125–9135. doi: 10.1088/1751-8113/40/30/030
  • J. Hong, S. Jiang, and C. Li, Explicit multi-symplectic methods for Klein–Gordon–Schrödinger equations, J. Comput. Phys. 228 (2009), pp. 3517–3532. doi: 10.1016/j.jcp.2009.02.006
  • L. Kong, R. Liu, and Z. Xu, Numerical simulation of interaction between Schrödinger field and Klein–Gordon field by multisymplectic method, Appl. Math. Comput. 181 (2006), pp. 342–350.
  • L. Kong, L. Wang, S. Jiang, and Y. Duan, Multisymplectic Fourier pseudo-spectral integrators for Klein–Gordon-Schrödinger equations, Sci. China Math. 56 (2013), pp. 915–932. doi: 10.1007/s11425-013-4575-3
  • V. Makhankov, Dynamics of classical solitons (in non-integrable systems), Phys. Lett. C 35 (1978), pp. 1–12.
  • J. Marsden, G. Patrick, and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Commun. Math. Phys. 199 (1999), pp. 351–395. doi: 10.1007/s002200050505
  • T. Ozawa and Y. Tsutsumi, Asymptotic behaviour of solutions for the coupled Klein–Gordon–Schrödinger equations, Adv. Stud. Pure Math. 23 (1994), pp. 295–305.
  • M. Ohta, Stability of stationary states for the coupled Klein–Gordon–Schrödinger equations, Nonlinear Anal. 27 (1996), pp. 455–461. doi: 10.1016/0362-546X(95)00017-P
  • A. Samarskii and V. Andreev, Difference Methods for Elliptic Equation, Nauka, Moscow, 1976.
  • J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.
  • J. Shen, T. Tang, and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Berlin, 2011.
  • Z. Sun and Q. Zhu, On Tsertsvadzes difference scheme for the Kuramoto-Tsuzuki equation, J. Comput. Appl. Math. 98 (1998), pp. 289–304. doi: 10.1016/S0377-0427(98)00135-6
  • A. Taleei and M. Dehghan, The solitary wave solution of coupled Klein–Gordon–Zakharov equations via two different numerical methods, Comput. Phys. Comm. 184 (2013), pp. 2145–2158. doi: 10.1016/j.cpc.2013.04.010
  • A. Taleei and M. Dehghan, A pseudo-spectral method that uses an overlapping multidomain technique for the numerical solution of sine-Gordon equation in one and two spatial dimensions, Math. Method Appl. Sci. 37 (2014), pp. 1909–1923. doi: 10.1002/mma.2943
  • A. Taleei and M. Dehghan, Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one and multi-dimensional nonlinear Schrödinger equations, Comput. Phys. Comm. 185 (2014), pp. 1515–1528. doi: 10.1016/j.cpc.2014.01.013
  • T. Wang, Optimal point-wise error estimate of a compact difference scheme for the Klein–Gordon–Schrödinger equation, J. Math. Anal. Appl. 412 (2014), pp. 155–167. doi: 10.1016/j.jmaa.2013.10.038
  • T. Wang and Y. Jiang, Point-wise errors of two conservative difference schemes for the Klein–Gordon–Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), pp. 4565–4575. doi: 10.1016/j.cnsns.2012.03.032
  • Y. Wang, B. Wang, and M. Qin, Local structure-preserving algorithms for partial differential equations, Sci. China Ser. A 51 (2008), pp. 2115–2136. doi: 10.1007/s11425-008-0046-7
  • S. Wang and L. Zhang, A class of conservative orthogonal spline collocation schemes for solving coupled Klein–Gordon-Schrödinger equations, Appl. Math. Comput. 203 (2008), pp. 799–812.
  • M. Wang and Y. Zhou, The periodic wave solutions for the Klein–Gordon-Schrödinger equations, Phys. Lett. A 318 (2003), pp. 84–92. doi: 10.1016/j.physleta.2003.07.026
  • J. Xia, S. Han, and M. Wang, The exact solitary wave solution for the Klein–Gordon–Schrödinger, Appl. Math. Mech. 23 (2002), pp. 52–58.
  • J. Xia and M. Wang, Exact solitary solution of coupled Klein–Gordon–Schrödinger equations, Appl. Math. Mech. 23 (2002), pp. 52–57.
  • X. Xiang, Spectral method for solving the system of equations of Schrödinger–Klein–Gordon field, J. Comput. Appl. Math. 21 (1988), pp. 161–171. doi: 10.1016/0377-0427(88)90265-8
  • L. Zhang, Convergence and stability of a conservative finite difference scheme for a class of equation system in interaction of complex Schrödinger field and real Klein–Gordon field, Numer. Math. J. Chinese Univ. 22 (2000), pp. 362–370.
  • L. Zhang, Convergence of a conservative difference schemes for a class of Klein–Gordon–Schrödinger equations in one space dimension, Appl. Math. Comput. 163 (2005), pp. 343–355.
  • J. Zhang and L. Kong, New energy-preserving schemes for Klein–Gordon–Schrödinger equations, Appl. Math. Model. 40 (2016), pp. 6969–6982. doi: 10.1016/j.apm.2016.02.026
  • Y. Zhou, Application of Discrete Functional Analysis to the Finite Difference Method, Inter. Acad. Publishers, Beijing, 1990.

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