Numerical analysis of a new conservative scheme for the coupled nonlinear Schrödinger equations

References

• G.D. Akrivis, V.A. Dougalis, and O.A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math. 59 (1991), pp. 31–53.
   Web of Science ®Google Scholar
• A. Aydin and B. Karasözen, Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions, Comput. Phys. Comm. 177 (2007), pp. 566–583.
   Web of Science ®Google Scholar
• D.J. Benney and A.C. Newell, The propagation of nonlinear wave envelopes, J. Math. Phys. 46 (1967), pp. 133–139.
   Web of Science ®Google Scholar
• C. Besse, B. Bidegaray, and S. Descombes, Order estimates in time of splitting methods for the nonlinear Schrödinger equation, SIAM J. Numer. Anal. 40 (2002), pp. 26–40.
   Web of Science ®Google Scholar
• H.P. Bhatt and A.Q.M. Khaliq, Higher order exponential time differencing scheme for system of coupled nonlinear Schrödinger equations, Appl. Math. Comput. 228 (2014), pp. 271–291.
   Web of Science ®Google Scholar
• T.J. Bridges and S. Reich, Multi-symplectic integrators: Numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A 284 (2001), pp. 184–193.
   Web of Science ®Google Scholar
• C. Canuto, M.Y. Hussaini, A. Quarteroni, M.Y. Hussaini, and T.A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006.
   Google Scholar
• C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comput. 38 (1982), pp. 67–86.
   Web of Science ®Google Scholar
• J.B. Chen and M.Z. Qin, Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation, Electron. Trans. Numer. Anal 12 (2001), pp. 193–204.
   Google Scholar
• Y.M. Chen, H.J. Zhu, and S.H. Song, Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation, Comput. Phys. Comm. 181 (2010), pp. 1231–1241.
   Web of Science ®Google Scholar
• Y.Z. Gong, J.X. Cai, and Y.S. Wang, Multi-symplectic Fourier pseudospectral method for the Kawahara equation, Commun. Comput. Phys. 16 (2014), pp. 35–55.
   Web of Science ®Google Scholar
• Y.Z. Gong and Y.S. Wang, Analysis of a new conservative Fourier pseudospectral algorithm for the coupled nonlinear Schrödinger equations, preprint.
   Google Scholar
• W.L. Guo, Zh.Q. Zhang, and H.P. Ma, Optimal error estimates of multi-symplectic method for nonlinear Schrödinger equation, J. Shanghai Univ. 15 (2009), pp. 487–492.
   Google Scholar
• P. Hansen, J.G. Nagy, and D.P. ÓLeary, Deblurring Images: Matrices, Spectra, and Filtering, Chapter 4, SIAM, Philadelphia, 2006.
   Google Scholar
• M.S. Ismail, Numerical solution of coupled nonlinear Schrödinger equation by Galerkin method, Math. Comput. Simulation 78 (2008), pp. 532–547.
   Web of Science ®Google Scholar
• M.S. Ismail and S.Z. Alamri, Highly accurate finite difference method for coupled nonlinear Schrödinger equation, Int. J. Comput. Math. 81 (2004), pp. 333–351.
   Web of Science ®Google Scholar
• L.H. Kong, J.L. Hong, L.H. Ji, and P.F. Zhu, Compact and efficient conservative schemes for coupled nonlinear Schrödinger equations, Numer. Methods Partial Differential Equations 31 (2015), pp. 1814–1843.
   Web of Science ®Google Scholar
• Y.P. Ma, L.H. Kong, J.L. Hong, and Y. Cao, High-order compact splitting multi-symplectic method for the coupled nonlinear Schrödinger equations, Comput. Math. Appl. 61 (2011), pp. 319–333.
   Web of Science ®Google Scholar
• J.E. Marsden, G.W. Patrick, and S. Shkoller, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys. 199 (1999), pp. 351–395.
   Web of Science ®Google Scholar
• J. Shen, T. Tang, and L.L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Berlin, 2011.
   Google Scholar
• W.J. Sonnier and C.I. Christov, Strong coupling of Schrödinger equations: Conservative scheme approach, Math. Comput. Simulation 69 (2005), pp. 514–525.
   Web of Science ®Google Scholar
• J.Q. Sun and M.Z. Qin, Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system, Comput. Phys. Comm. 155 (2003), pp. 221–235.
   Web of Science ®Google Scholar
• Z.Z. Sun and D.D. Zhao, On the L∞ convergence of a difference scheme for coupled nonlinear Schrödinger equations, Comput. Math. Appl. 59 (2010), pp. 3286–3300.
   Web of Science ®Google Scholar
• M. Thalhammer, Convergence analysis of high-order time-splitting pseudospectral methods for nonlinear Schrödinger equations, SIAM J. Numer. Anal. 50 (2012), pp. 3231–3258.
   Web of Science ®Google Scholar
• T.C. Wang, Optimal point-wise error estimate of a compact finite difference scheme for the coupled nonlinear Schrödinger equations, J. Comput. Math. 32 (2014), pp. 58–74.
   Web of Science ®Google Scholar
• T.C. Wang, B.L. Guo, and Q.B. Xu, Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions, J. Comput. Phys. 243 (2013), pp. 382–399.
   Web of Science ®Google Scholar
• Y.S. Wang and S.T. Li, New schemes for the coupled nonlinear Schrödinger equation, Int. J. Comput. Math. 87 (2010), pp. 775–787.
   Web of Science ®Google Scholar
• Y.S. Wang, B. Wang, and M.Z. Qin, Local structure-preserving algorithms for partial differential equations, Sci. China Ser. A 51 (2008), pp. 2115–2136.
   Google Scholar
• Y.L. Zhou, Applications of Discrete Functional Analysis to the Finite Difference Method, International Academic Publishers, Beijing, 1990.
   Google Scholar