Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-15T05:45:56.107Z Has data issue: false hasContentIssue false

Novel Conformal Structure-Preserving Algorithms for Coupled Damped Nonlinear Schrödinger System

Published online by Cambridge University Press:  28 November 2017

Xu Qian*
Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan 410072, China
Songhe Song
Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan 410072, China State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, Hunan 410073, China
*Corresponding author. (X. Qian)
Get access


This paper introduces two novel conformal structure-preserving algorithms for solving the coupled damped nonlinear Schrödinger (CDNLS) system, which are based on the conformal multi-symplectic Hamiltonian formulation and its conformal conservation laws. The proposed algorithms can preserve corresponding conformal multi-symplectic conservation law and conformal momentum conservation law in any local time-space region, respectively. Moreover, it is further shown that the algorithms admit the conformal charge conservation law, and exactly preserve the dissipation rate of charge under appropriate boundary conditions. Numerical experiments are presented to demonstrate the conformal properties and effectiveness of the proposed algorithms during long-time numerical simulations and validate the analysis.

Research Article
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1] Abo-Shaeer, J. R., Raman, C., Vogels, J. M. and Ketterle, W., Observation of vortex lattices in Bose-Einstein condensates, Science, 292 (2001), pp. 476479.CrossRefGoogle ScholarPubMed
[2] Correggi, M. and Yngvason, J., Energy and vorticity in fast rotating Boseš-Einstein condensates, J. Phys. A Math. Theor., 41 (2008), 445002.CrossRefGoogle Scholar
[3] Hall, D. S., Matthews, M. R., Ensher, J. R., Wieman, C. E. and Cornell, E. A., Dynamics of component separation in a binary mixture of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 1539.CrossRefGoogle Scholar
[4] Benney, D. J. and Newell, A. C., The propagation of nonlinear wave envelopes, J. Math. Phys., 46 (1967), pp. 133139.CrossRefGoogle Scholar
[5] Madison, K. W., Chevy, F., Wohlleben, W. and Dalibard, J., Vortices in a stirred Bose-Einstein condensate, J. Mod. Opt., 47 (2000), pp. 27152723.CrossRefGoogle Scholar
[6] Lin, T. C. and Wei, J., Ground state of N coupled nonlinear Schrödinger equations in Rn, n≤3, Commun. Math. Phys., 255 (2005), pp. 629653.CrossRefGoogle Scholar
[7] Liu, Z., Phase separation of two-component Boseš-Einstein condensates, J. Math. Phys., 50 (2009), 102104.CrossRefGoogle Scholar
[8] Fushchych, W. I. and Symenoh, Z. I., Symmetry of equations with convection terms, J. Nonlinear Math. Phys., 4 (1997), pp. 470479.CrossRefGoogle Scholar
[9] Bandrauk, A. D. and Shen, H., High-order split-step exponential methods for solving coupled nonlinear Schrödinger equations, J. Phys. A Math. Gen., 27 (1994), 7147.CrossRefGoogle Scholar
[10] Ismail, M. S. and Taha, T. R., Numerical simulation of coupled nonlinear Schrödinger equation, Math. Comput. Simul., 56 (2001), pp. 547562.CrossRefGoogle Scholar
[11] Bridges, T. J. and Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), pp. 184193.CrossRefGoogle Scholar
[12] Bridges, T. J. and Reich, S., Multi-symplectic spectral discretizations for the Zakharovš- Kuznetsov and shallow water equations, Phys. D, 152 (2001), pp. 491504.CrossRefGoogle Scholar
[13] Hong, J. L., Liu, X. Y. and Li, C., Multi-symplectic Rungeš-Kuttaš-Nyström methods for nonlinear Schrödinger equations with variable coefficients, J. Comput. Phys., 226 (2007), pp. 19681984.CrossRefGoogle Scholar
[14] Sun, Y. J. and Tse, P. S. P., Symplectic and multisymplectic numerical methods for Maxwell equations, J. Comput. Phys., 230 (2011), pp. 20762094.CrossRefGoogle Scholar
[15] Li, H. C., Sun, J. Q. and Qin, M. Z., New explicit multi-symplectic scheme for nonlinear wave equation, Appl. Math. Mech. Engl. Ed., 35 (2014), pp. 369380.CrossRefGoogle Scholar
[16] Qin, Y. Y., Deng, Z. C. and Hu, W. P., Structure-preserving properties of three differential schemes for oscillator system, Appl. Math. Mech. Engl. Ed., 35 (2014), pp. 783790.CrossRefGoogle Scholar
[17] Sun, J. Q. and Qin, M. Z., Multi-symplectic methods for the coupled 1D nonlinear Schrödinger system, Comput. Phys. Commun., 155 (2003), pp. 221235.CrossRefGoogle Scholar
[18] Aydin, A. and Karasözen, B., Multi-symplectic integration of coupled non-linear Schrödinger system with soliton solutions, Int. J. Comput. Math., 86 (2009), pp. 864882.CrossRefGoogle Scholar
[19] 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), pp. 12311241.CrossRefGoogle Scholar
[20] Ma, Y. P., Kong, L. H., Hong, J. L. and Cao, Y., High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations, Comput. Math. Appl., 61 (2011), pp. 319333.CrossRefGoogle Scholar
[21] Qian, X., Song, S. H. and Chen, Y. M., A semi-explicit multi-symplectic splitting scheme for a 3-coupled nonlinear Schrödinger equation, Comput. Phys. Commun., 185 (2014), pp. 12551264.CrossRefGoogle Scholar
[22] Wang, Y. S., Wang, B. and Qin, M. Z., Local structure-preserving algorithms for partial differential equations, Sci. Chins Ser. A Math., 51 (2008), pp. 21152136.CrossRefGoogle Scholar
[23] Cai, J. X. and Wang, Y. S., Local structure-preserving algorithms for the “good” Boussinesq equation, J. Comput. Phys., 239 (2013), pp. 7289.CrossRefGoogle Scholar
[24] Cai, J. X., Wang, Y. S. and Liang, H., Local energy-preserving and momentum-preserving algorithms for coupled nonlinear Schrödinger system, J. Comput. Phys., 239 (2013), pp. 3050.CrossRefGoogle Scholar
[25] Gong, Y. Z., Cai, J. X. and Wang, Y. S., Some new structure-preserving algorithms for general multi-symplectic formulations of Hamiltonian PDEs, J. Comput. Phys., 279 (2014), pp. 80102.CrossRefGoogle Scholar
[26] Li, Y. W. and Wu, X. Y., General local energy-preserving integrators for solving multi-symplectic Hamiltonian PDEs, J. Comput. Phys., 301 (2015), pp. 141166.CrossRefGoogle Scholar
[27] McLachlan, R. and Perlmutter, M., Conformal Hamiltonian systems, J. Geom. Phys., 39 (2001), pp. 276300.CrossRefGoogle Scholar
[28] McLachlan, R. I. and Quispel, G. R. W., Splitting methods, Acta Numer., 11 (2002), pp. 341434.CrossRefGoogle Scholar
[29] Moore, B. E., Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations, Math. Comput. Simulation, 80 (2009), pp. 2028.CrossRefGoogle Scholar
[30] Moore, B. E., Noreña, L. and Schober, C. M., Conformal conservation laws and geometric integration for damped Hamiltonian PDEs, J. Comput. Phys., 232 (2013), pp. 214233.CrossRefGoogle Scholar
[31] Fu, H., Zhou, W. E., Qian, X., Song, S. H. and Zhang, L. Y., Conformal structure-preserving method for damped nonlinear Schrödinger equation, China Phys. B, 25 (2016), 110201.CrossRefGoogle Scholar