{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T12:08:11Z","timestamp":1759838891944,"version":"3.37.3"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T00:00:00Z","timestamp":1556668800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771438"],"award-info":[{"award-number":["11771438"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2019,8]]},"DOI":"10.1007\/s11075-019-00708-8","type":"journal-article","created":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T09:02:35Z","timestamp":1556701355000},"page":"1485-1503","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Symplectic simulation of dark solitons motion for nonlinear Schr\u00f6dinger equation"],"prefix":"10.1007","volume":"81","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8775-0108","authenticated-orcid":false,"given":"Beibei","family":"Zhu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yifa","family":"Tang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ruili","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yihao","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,5,1]]},"reference":[{"issue":"6","key":"708_CR1","doi-asserted-by":"publisher","first-page":"1011","DOI":"10.1063\/1.523009","volume":"17","author":"MJ Ablowitz","year":"1976","unstructured":"Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equations and fourier analysis. J. Math. Phys. 17(6), 1011\u20131018 (1976)","journal-title":"J. Math. Phys."},{"key":"708_CR2","volume-title":"Foundations of mechanics","author":"RE Abraham","year":"1978","unstructured":"Abraham, R.E., Marsden, J.E.: Foundations of mechanics. Benjamin-Cummings, Reading (1978)"},{"key":"708_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1693-1","volume-title":"Mathematical methods of classical mechanics","author":"VI Arnold","year":"1978","unstructured":"Arnold, V.I.: Mathematical methods of classical mechanics. Springer, New York (1978)"},{"key":"708_CR4","first-page":"3","volume":"318","author":"L Barletti","year":"2018","unstructured":"Barletti, L., Brugnano, L., Frasca Caccia, G., Iavernaro, F.: Energy-conserving methods for the nonlinear Schr\u00f6dinger equation. Appl. Math. Comput. 318, 3\u201318 (2018)","journal-title":"Appl. Math. Comput."},{"key":"708_CR5","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1016\/S0377-0427(01)00492-7","volume":"142","author":"S Blanes","year":"2002","unstructured":"Blanes, S., Moan, P.C.: Practical symplectic Runge-Kutta and Runge-Kutta-Nystr\u00f6m methods. J. Comput. Appl. Math. 142, 313\u2013330 (2002)","journal-title":"J. Comput. Appl. Math."},{"key":"708_CR6","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/j.cnsns.2017.12.018","volume":"60","author":"L Brugnano","year":"2018","unstructured":"Brugnano, L., Zhang, C.J., Li, D.F.: A class of energy-conserving Hamiltonian boundary value methods for nonlinear Schr\u00f6dinger equation with wave operator. Commun. Nonlinear Sci. Numer. Simulat. 60, 33\u201349 (2018)","journal-title":"Commun. Nonlinear Sci. Numer. Simulat."},{"key":"708_CR7","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1090\/S0025-5718-1964-0159424-9","volume":"18","author":"JC Butcher","year":"1964","unstructured":"Butcher, J.C.: Implicit Runge-Kutta processes. Math. Comput. 18, 50\u201364 (1964)","journal-title":"Math. Comput."},{"key":"708_CR8","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1016\/j.jcp.2013.01.009","volume":"239","author":"JX Cai","year":"2013","unstructured":"Cai, J.X., Wang, Y.S.: Local structure-preserving algorithms for the \u201cgood\u201d Boussinesq equation. J. Comput. Phys. 239, 72\u201389 (2013)","journal-title":"J. Comput. Phys."},{"key":"708_CR9","doi-asserted-by":"publisher","first-page":"1689","DOI":"10.1090\/S0025-5718-2014-02805-6","volume":"83","author":"E Celledoni","year":"2014","unstructured":"Celledoni, E., Owren, B., Sun, Y.: The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the Averaged Vector Field method. Math. Comp. 83, 1689\u20131700 (2014)","journal-title":"Math. Comp."},{"issue":"2","key":"708_CR10","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1088\/0951-7715\/3\/2\/001","volume":"3","author":"PJ Channell","year":"1990","unstructured":"Channell, P.J., Scovel, J.C.: Symplectic integration of hamiltonian systems. Nonlinearity 3(2), 231\u2013259 (1990)","journal-title":"Nonlinearity"},{"key":"708_CR11","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1093\/imanum\/7.1.1","volume":"7","author":"GJ Cooper","year":"1987","unstructured":"Cooper, G.J.: Stability of Runge-Kutta Methods for Trajectory Problems. IMA J. Numer. Anal. 7, 1\u201313 (1987)","journal-title":"IMA J. Numer. Anal."},{"key":"708_CR12","unstructured":"Dodd, R.K., Eibeck, J.C., Gibbon, J.D., Morris, H.: Solitons and nonlinear wave equation. Academic Press (1982)"},{"key":"708_CR13","unstructured":"Feng, K.: On difference schemes and symplectic geometry Feng, K. (ed.) . Science Press, Beijing (1985)"},{"issue":"1","key":"708_CR14","first-page":"88","volume":"6","author":"Z Ge","year":"1988","unstructured":"Ge, Z., Feng, K.: On the approximation of linear Hamiltonian systems. J. Computa. Math. 6(1), 88\u201397 (1988)","journal-title":"J. Computa. Math."},{"key":"708_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-05018-7","volume-title":"Geometric numerical integration","author":"E Hairer","year":"2002","unstructured":"Hairer, E., Lubich, C.h., Wanner, G.: Geometric numerical integration. Springer, New York (2002)"},{"key":"708_CR16","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0108677","volume-title":"Optical solitons in fibers","author":"A Hasegawa","year":"1989","unstructured":"Hasegawa, A.: Optical solitons in fibers. Springer-Verlag, Berlin (1989)"},{"key":"708_CR17","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1016\/0378-4754(94)00024-7","volume":"37","author":"BM Herbst","year":"1994","unstructured":"Herbst, B.M., Varadi, F., Ablowitz, M.J.: Symplectic methods for the nonlinear Schr\u00f6dinger equation. Math. Comput. Simul. 37, 353\u2013369 (1994)","journal-title":"Math. Comput. Simul."},{"key":"708_CR18","doi-asserted-by":"publisher","DOI":"10.1142\/2320","volume-title":"Nonlinear random waves","author":"VV Konotop","year":"1994","unstructured":"Konotop, V.V., V\u00e1zquez, L.: Nonlinear random waves. World Scientific, Singapore (1994)"},{"key":"708_CR19","doi-asserted-by":"publisher","first-page":"767","DOI":"10.1088\/0305-4470\/24\/4\/013","volume":"24","author":"VV Konotop","year":"1991","unstructured":"Konotop, V.V., Vekslerchik, V.E.: Randomly modulated dark soliton. J. Phys. A: Math. Gen. 24, 767\u2013785 (1991)","journal-title":"J. Phys. A: Math. Gen."},{"key":"708_CR20","unstructured":"Konotop, V.V., Tang, Y.F.: Personal communication (1996)"},{"key":"708_CR21","doi-asserted-by":"publisher","first-page":"2399","DOI":"10.1103\/PhysRevLett.81.2399","volume":"81","author":"RI McLachlan","year":"1998","unstructured":"McLachlan, R.I., Quispel, G.R., Robidoux, N.: Unified approach to Hamiltonian systems, poisson systems, gradient systems, and systems with Lyapunov functions or first integrals. Phys. Rev. Lett. 81, 2399\u20132403 (1998)","journal-title":"Phys. Rev. Lett."},{"key":"708_CR22","doi-asserted-by":"publisher","first-page":"1021","DOI":"10.1098\/rsta.1999.0363","volume":"357","author":"RI McLachlan","year":"1999","unstructured":"McLachlan, R.I., Quispel, G.R.W., Robidoux, N.: Geometric integration using discrete gradients. Phil. Trans. Roy. Soc. A 357, 1021\u20131045 (1999)","journal-title":"Phil. Trans. Roy. Soc. A"},{"issue":"2","key":"708_CR23","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1137\/0141018","volume":"41","author":"JW Miles","year":"1981","unstructured":"Miles, J.W.: An envelope soliton problem. SIAM J. Appl. Math. 41(2), 227\u2013230 (1981)","journal-title":"SIAM J. Appl. Math."},{"key":"708_CR24","doi-asserted-by":"publisher","first-page":"045206","DOI":"10.1088\/1751-8113\/41\/4\/045206","volume":"41","author":"GRW Quispel","year":"2008","unstructured":"Quispel, G.R.W., McLaren, D.I.: A new class of energy-preserving numerical integration methods. J. Phys. A: Math. Theor. 41, 045206 (2008)","journal-title":"J. Phys. A: Math. Theor."},{"key":"708_CR25","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1007\/BF01954907","volume":"28","author":"JM Sanz-Serna","year":"1988","unstructured":"Sanz-Serna, J.M.: Runge-Kutta schemes for Hamiltonian systems. BIT Numer. Math. 28, 877\u2013883 (1988)","journal-title":"BIT Numer. Math."},{"key":"708_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3093-4","volume-title":"Numerical Hamiltonian problems","author":"JM Sanz-Serna","year":"1994","unstructured":"Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian problems. Chapman & Hall, London (1994)"},{"key":"708_CR27","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1016\/S0375-9601(99)00353-9","volume":"259","author":"CM Schober","year":"1999","unstructured":"Schober, C.M.: Symplectic integrators for the Ablowitz-Ladik discrete nonlinear Schr\u00f6dinger equation. Phys. Lett. A 259, 140\u2013151 (1999)","journal-title":"Phys. Lett. A"},{"key":"708_CR28","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1137\/0705041","volume":"5","author":"G Strang","year":"1968","unstructured":"Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 507\u2013517 (1968)","journal-title":"SIAM J. Numer. Anal."},{"issue":"10","key":"708_CR29","doi-asserted-by":"publisher","first-page":"2425","DOI":"10.1088\/1751-8113\/40\/10\/012","volume":"40","author":"YF Tang","year":"2007","unstructured":"Tang, Y.F., Cao, J.W., Liu, X.T., Sun, Y.C.: Symplectic methods for Ablowitz-Ladik discrete nonlinear Schr\u00f6dinger equation. J. Phys. A: Math. Theor. 40(10), 2425\u20132437 (2007)","journal-title":"J. Phys. A: Math. Theor."},{"key":"708_CR30","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1016\/S0096-3003(96)00019-7","volume":"82","author":"YF Tang","year":"1997","unstructured":"Tang, Y.F., P\u00e9rez-Garc\u00eda V.M., V\u00e1zquez, L.: Symplectic methods for the Ablowitz-Ladik model. Appli. Math. Comput. 82, 17\u201338 (1997)","journal-title":"Appli. Math. Comput."},{"issue":"5","key":"708_CR31","first-page":"823","volume":"37","author":"VE Zakharov","year":"1973","unstructured":"Zakharov, V.E., Shabat, A.B.: Interaction between solitions in a stable medium. Sov. Phys.-JETP 37(5), 823\u2013828 (1973)","journal-title":"Sov. Phys.-JETP"},{"issue":"2-3","key":"708_CR32","first-page":"165","volume":"71","author":"F Zhang","year":"1995","unstructured":"Zhang, F., P\u00e9rez-Garc\u00eda V.M., V\u00e1zquez, L.: Numerical simulation of nonlinear Schr\u00f6dinger systems: A new conservative scheme. Appl. Math. Comput. 71(2-3), 165\u2013177 (1995)","journal-title":"Appl. Math. Comput."},{"key":"708_CR33","unstructured":"Zhang, R.L., Huang, J.F., Tang, Y.F., V\u00e1zquez, L.: Revertible and symplectic methods for the Ablowitz-Ladik discrete nonlinear Schr\u00f6dinger equation. In: Proceedings of the 2011 summer simulation multiconference (27\u201329 June \u2014 The Hague, Netherlands): Grand Challenges in Modeling and Simulation (GCMS\u201911), ISBN: 1-56555-345-4, The Society for Modeling and Simulation International (SCS), San Diego USA (2011)"},{"key":"708_CR34","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1063\/1.4982743","volume":"24","author":"ZQ Zhou","year":"2017","unstructured":"Zhou, Z.Q., He, Y., Sun, Y.J., Liu, J., Qin, H.: Explicit symplectic methods for solving charged particle trajectories. Phys. Plasmas 24, 87\u201394 (2017)","journal-title":"Phys. Plasmas"},{"issue":"10","key":"708_CR35","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1016\/j.jcp.2016.06.044","volume":"322","author":"BB Zhu","year":"2016","unstructured":"Zhu, B.B., Zhang, R.L., Tang, Y.F., Tu, X.B., Zhao, Y.: Splitting K-symplectic methods for non-canonical separable Hamiltonian problems. J. Computa. Phys. 322(10), 387\u2013399 (2016)","journal-title":"J. Computa. Phys."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00708-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00708-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00708-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,4,30]],"date-time":"2020-04-30T00:04:14Z","timestamp":1588205054000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00708-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,1]]},"references-count":35,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,8]]}},"alternative-id":["708"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00708-8","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2019,5,1]]},"assertion":[{"value":"31 May 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 May 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}