Skip to main content Accessibility help

Numerical Solutions of Coupled Nonlinear Schrödinger Equations by Orthogonal Spline Collocation Method

  • Qing-Jiang Meng (a1), Li-Ping Yin (a2), Xiao-Qing Jin (a1) and Fang-Li Qiao (a3)


In this paper, we present the use of the orthogonal spline collocation method for the semi-discretization scheme of the one-dimensional coupled nonlinear Schrödinger equations. This method uses the Hermite basis functions, by which physical quantities are approximated with their values and derivatives associated with Gaussian points. The convergence rate with order and the stability of the scheme are proved. Conservation properties are shown in both theory and practice. Extensive numerical experiments are presented to validate the numerical study under consideration.


Corresponding author

Corresponding author.Email


Hide All
[1]Ablowitz, M. J. and Segur, H., Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.
[2]Antoine, X., Arnold, A., Besse, C., Ehrhardt, M. and Schadle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., 4 (2008), 729796.
[3]Bao, W. Z., Ground states and dynamics of multicomponent Bose-Einstein condensates, Multiscale Model. Simul., 2 (2004), 201236.
[4]Bao, W. Z. and Shen, J., A fourth-order time-splitting Laguerre-Hermite pseudospectral method for Bose-Einstein condensates, SIAM J. Sci. Comput., 26 (2005), 20102028.
[5]Bao, W. Z. and Zheng, C. X., A time-splitting spectral method for three-wave interactions in media with competing quadratic and cubic nonlinearities, Commun. Comput. Phys., 2 (2007), 123140.
[6]Benney, D. J. and Newell, A. C., Random wave closures, Stud. Appl. Math., 48 (1969), 2953.
[7]de Boor, C. and Swartz, B., Collocation at Gauss points, SIAM. J. Numer. Anal., 10 (1973), 582606.
[8]Chang, Q. S., Jia, E. H. and Sun, W. W., Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), 397415.
[9]Chen, Y. M., Zhu, H. J. and Song, S. H., Multi-symplectic splitting method for the coupled nonlinear Schrödinger equation, Comput. Phys. Commun., 181 (2010), 12311241.
[10]Douglas, J. Jr and Dupont, T., Collocation Methods for Parabolic Equations in a Single Space Variable, Lecture Notes in Math, Vol. 385, Spring-Verleg, New York, 1974.
[11]Fairweather, G. and Meade, D., A survey of spline collocation methods for the numerical solution of differential equations, in: Diaz, J. C. ed., Mathematics for Large Scale Computing, Lecture Notes in Pure Appl. Math., Vol. 120, Marcel Dekker, New York, (1989), 297341.
[12]Guo, B. Y., Pedro, J. P., Maria, J. R. and Luis, V., Numerical solution of the Sine-Gordon equation, Appl. Math. Comput., 18 (1986), 114.
[13]Ismail, M. S. and Alamri, S. Z., Highly accurate finite difference method for coupled nonlinear Schrödinger equation, Int. J. Comput. Math., 81 (2004), 333351.
[14]Ismail, M. S. and Taha, T. R., Numerical simulation of coupled nonlinear Schrödinger equation, Math. Comput. Simul., 56 (2001), 547562.
[15]Klein, P., Antoine, X., Besse, C. and Ehrhardt, M., Absorbing boundary conditions for solving N-dimensional stationary Schrödinger equations with unbounded potentials and nonlinearities, Commun. Comput. Phys., 10 (2011), 12801304.
[16]Menyuk, C. R., Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron., 23 (1987), 174176.
[17]Robinson, M. P. and Fairweather, G., Orthogonal spline collocation methods for Schrödinger-type equations in one space variable, Numer. Math., 68 (1994), 355376.
[18]Sun, J. Q., Gu, X. Y. and Ma, Z. Q., Numerical study for the soliton waves of the coupled nonlinear Schrödinger system, Phys. D, 196 (2004), 311328.
[19]Sun, J. Q. and Qin, M. Z., Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system, Comput. Phys. Commun., 155 (2003), 221235.
[20]Thalhammer, M., Caliari, M. and Neuhauser, C., High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822832.
[21]Utsumi, T., Aoki, T., Koga, J. and Yamagiwa, M., Solutions of the 1D coupled nonlinear schrödinger equations by the CIP-BS method, Commun. Comput. Phys., 1 (2006), 261275.
[22]Ueda, T. and Kath, W. L., Dynamics of coupled solitons in nonlinear optical fibers, Phys. Rev. A, 42 (1990), 563571.
[23]Wang, H. Q., Numerical studies on the split step finite difference method for the nonlinear Schrödinger equations, Appl. Math. Comput., 170 (2005), 1735.
[24]Wadati, M., Izuka, T. and Hisakado, M., A coupled nonlinear Schrödinger equation and optical solitons, J. Phys. Soc. Jpn., 61 (1992), 22412245.
[25]Xu, Y. and Shu, C. W., Local discontinuous Galerkin methods for nonlinear Schrödinger equations, J. Comput. Phys., 205 (2005), 7297.
[26]Xu, Y. and Shu, C. W., Local discontinuous Galerkin methods for high-order time-dependent partial differential equations, Commun. Comput. Phys., 7 (2010), 146.
[27]Xu, Y. and Shu, C. W., Local discontinuous Galerkin methods for the Degasperis-Procesi equation, Commun. Comput. Phys., 10 (2011), 474508.
[28]Yang, J. K., Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics, Phys. Rev. E, 59 (1999), 23932405.
[29]Yang, J. K. and Benney, D. J., Some properties of nonlinear wave systems, Stud. Appl. Math., 96 (1996), 111135.
[30]Zhang, Y. Z., Bao, W. Z. and Li, H. L., Dynamics of rotating two-component Bose-Einstein condensates and its efficient computation, Phys. D, 234 (2007), 4969.


Numerical Solutions of Coupled Nonlinear Schrödinger Equations by Orthogonal Spline Collocation Method

  • Qing-Jiang Meng (a1), Li-Ping Yin (a2), Xiao-Qing Jin (a1) and Fang-Li Qiao (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed