{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T11:03:13Z","timestamp":1756810993336,"version":"3.37.3"},"reference-count":56,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,2,10]],"date-time":"2024-02-10T00:00:00Z","timestamp":1707523200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,2,10]],"date-time":"2024-02-10T00:00:00Z","timestamp":1707523200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12301516"],"award-info":[{"award-number":["12301516"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Key International Cooperative Research Projects of the National Natural Science Foundation of China","award":["12120101001"],"award-info":[{"award-number":["12120101001"]}]},{"name":"Major Basic Research Project of Natural Science Foundation of Shandong Province","award":["ZR2021ZD03"],"award-info":[{"award-number":["ZR2021ZD03"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2024,11]]},"DOI":"10.1007\/s11075-023-01745-0","type":"journal-article","created":{"date-parts":[[2024,2,10]],"date-time":"2024-02-10T11:02:10Z","timestamp":1707562930000},"page":"1179-1214","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Improved uniform error bounds of Lawson-type exponential integrator method for long-time dynamics of the high-dimensional space fractional sine-Gordon equation"],"prefix":"10.1007","volume":"97","author":[{"given":"Junqing","family":"Jia","sequence":"first","affiliation":[]},{"given":"Xiaoyun","family":"Jiang","sequence":"additional","affiliation":[]},{"given":"Xiaoqing","family":"Chi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,2,10]]},"reference":[{"issue":"4","key":"1745_CR1","doi-asserted-by":"crossref","first-page":"1689","DOI":"10.1137\/16M1075302","volume":"55","author":"M Ainsworth","year":"2017","unstructured":"Ainsworth, M., Mao, Z.: Analysis and approximation of a fractional Cahn-Hilliard equation. SIAM J. Numer. Anal. 55(4), 1689\u20131718 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"1745_CR2","doi-asserted-by":"crossref","first-page":"3056","DOI":"10.1137\/140954040","volume":"52","author":"Y Huang","year":"2014","unstructured":"Huang, Y., Oberman, A.: Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM J. Numer. Anal. 52, 3056\u20133084 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"1745_CR3","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/j.jcp.2019.05.048","volume":"395","author":"W Cai","year":"2019","unstructured":"Cai, W., Jiang, C., Wang, Y., Song, Y.: Structure-preserving algorithms for the two-dimensional sine-Gordon equation with Neumann boundary conditions. J. Comput. Phys. 395, 166\u2013185 (2019)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"1745_CR4","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/j.cam.2006.07.002","volume":"206","author":"AG Bratsos","year":"2007","unstructured":"Bratsos, A.G.: The solution of the two-dimensional sine-Gordon equation using the method of lines. J. Comput. Appl. Math. 206(1), 251\u2013277 (2007)","journal-title":"J. Comput. Appl. Math."},{"key":"1745_CR5","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1080\/00018736500101091","volume":"14","author":"JD Josephson","year":"1965","unstructured":"Josephson, J.D.: Supercurrents through barries. Adv. Phys. 14, 419\u2013451 (1965)","journal-title":"Adv. Phys."},{"key":"1745_CR6","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-06722-3","volume-title":"The sine-Gordon model and its applications","author":"J Cuevas-Maraver","year":"2014","unstructured":"Cuevas-Maraver, J., Kevrekidis, P.G., Williams, F.: The sine-Gordon model and its applications. Springer, Heidelberg (2014)"},{"key":"1745_CR7","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1016\/j.cpc.2009.12.010","volume":"181","author":"M Dehghan","year":"2010","unstructured":"Dehghan, M., Ghesmati, A.: Numerical simulation of two-dimensional sine-Gordon solitons via a local weak meshless technique based on the radial point interpolation method (RPIM). Comput. Phys. Commun. 181, 772\u2013786 (2010)","journal-title":"Comput. Phys. Commun."},{"key":"1745_CR8","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1016\/j.cma.2007.08.016","volume":"197","author":"M Dehghan","year":"2008","unstructured":"Dehghan, M., Mirzaei, D.: The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput. Methods Appl. Mech. Eng. 197, 476\u2013486 (2008)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1745_CR9","doi-asserted-by":"crossref","first-page":"2737","DOI":"10.1016\/j.cam.2009.11.022","volume":"233","author":"D Mirzaei","year":"2010","unstructured":"Mirzaei, D., Dehghan, M.: Meshless local Petrov-Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation. J. Comput. Appl. Math. 233, 2737\u20132754 (2010)","journal-title":"J. Comput. Appl. Math."},{"key":"1745_CR10","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1016\/j.physleta.2006.09.044","volume":"361","author":"H Susanto Darminto","year":"2007","unstructured":"Susanto Darminto, H., van Gils, S.A.: Static sand dynamic properties of fluxons in a zig-zag 0- $$\\pi $$ Josephson junction. Phys. Lett. A 361, 270\u2013276 (2007)","journal-title":"Phys. Lett. A"},{"key":"1745_CR11","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.enganabound.2014.09.008","volume":"50","author":"M Dehghan","year":"2015","unstructured":"Dehghan, M., Abbaszadeh, M., Mohebbi, A.: An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations. Eng. Anal. Bound. Elem. 50, 412\u2013434 (2015)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"1745_CR12","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/j.enganabound.2014.11.023","volume":"52","author":"M Ilati","year":"2015","unstructured":"Ilati, M., Dehghan, M.: The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations. Eng. Anal. Bound. Elem. 52, 99\u2013109 (2015)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"1745_CR13","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1002\/num.20289","volume":"24","author":"M Dehghan","year":"2008","unstructured":"Dehghan, M., Shokri, A.: A numerical method for one-dimensional nonlinear sine-Gordon equation using collocation and radial basis functions. Numer. Methods Partial Differential Equations 24, 687\u2013698 (2008)","journal-title":"Numer. Methods Partial Differential Equations"},{"key":"1745_CR14","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1016\/j.mcm.2009.11.015","volume":"51","author":"A Mohebbia","year":"2010","unstructured":"Mohebbia, A., Dehghan, M.: High-order solution of one-dimensional Sine-Gordon equation using compact finite difference and DIRKN methods. Math. Comput. Model. 51, 537\u2013549 (2010)","journal-title":"Math. Comput. Model."},{"key":"1745_CR15","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1016\/j.matcom.2005.02.017","volume":"68","author":"Q Sheng","year":"2005","unstructured":"Sheng, Q., Khaliq, A.Q.M., Voss, D.A.: Numerical simulation of two-dimensional sine-Gordon solitons via a split cosine scheme. Math. Comput. Simul. 68, 355\u2013373 (2005)","journal-title":"Math. Comput. Simul."},{"key":"1745_CR16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0045-7825(91)90136-T","volume":"86","author":"J Argyris","year":"1991","unstructured":"Argyris, J., Haase, M., Heinrich, J.C.: Finite element approximation to two-dimensional sine-Gordon solitons. Comput. Methods Appl. Mech. Eng. 86, 1\u201326 (1991)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1745_CR17","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.matcom.2008.04.018","volume":"79","author":"M Dehghan","year":"2008","unstructured":"Dehghan, M., Shokri, A.: A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions. Math. Comput. Simul. 79, 700\u2013715 (2008)","journal-title":"Math. Comput. Simul."},{"key":"1745_CR18","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/s00211-011-0411-2","volume":"120","author":"W Bao","year":"2012","unstructured":"Bao, W., Dong, X.: Analysis and comparison of numerical methods for the Klein-Gordon equation in the nonrelativistic limit regime. Numer. Math. 120, 189\u2013229 (2012)","journal-title":"Numer. Math."},{"key":"1745_CR19","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1006\/jcph.1993.1183","volume":"108","author":"W Cao","year":"1993","unstructured":"Cao, W., Guo, B.: Fourier collocation method for solving nonlinear Klein-Gordon equation. J. Comput. Phys. 108, 296\u2013305 (1993)","journal-title":"J. Comput. Phys."},{"key":"1745_CR20","doi-asserted-by":"crossref","first-page":"1909","DOI":"10.1002\/mma.2943","volume":"37","author":"A Taleei","year":"2014","unstructured":"Taleei, A., Dehghan, M.: 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. Methods Appl. Sci. 37, 1909\u20131923 (2014)","journal-title":"Math. Methods Appl. Sci."},{"key":"1745_CR21","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.1007\/s11425-020-1857-5","volume":"65","author":"W Cao","year":"2022","unstructured":"Cao, W., Li, D., Zhang, Z.: Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations. Sci China Math 65, 1731\u20131748 (2022)","journal-title":"Sci China Math"},{"key":"1745_CR22","doi-asserted-by":"crossref","first-page":"1307","DOI":"10.4208\/cicp.2019.js60.03","volume":"26","author":"W Bao","year":"2019","unstructured":"Bao, W., Feng, Y., Yi, W.: Long time error analysis of finite difference time domain methods for the nonlinear Klein-Gordon equation with weak nonlinearity. Commun. Comput. Phys. 26, 1307\u20131334 (2019)","journal-title":"Commun. Comput. Phys."},{"key":"1745_CR23","doi-asserted-by":"crossref","first-page":"1897","DOI":"10.1155\/S1073792804133321","volume":"37","author":"JM Delort","year":"2004","unstructured":"Delort, J.M., Szeftel, J.: Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres. Int. Math. Res. Not. 37, 1897\u20131966 (2004)","journal-title":"Int. Math. Res. Not."},{"key":"1745_CR24","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.jde.2010.03.025","volume":"249","author":"D Fang","year":"2010","unstructured":"Fang, D., Zhang, Q.: Long-time existence for semi-linear Klein-Gordon equations on tori. J. Differential Equations 249, 151\u2013179 (2010)","journal-title":"J. Differential Equations"},{"key":"1745_CR25","doi-asserted-by":"crossref","first-page":"1962","DOI":"10.1137\/21M1449774","volume":"60","author":"W Bao","year":"2022","unstructured":"Bao, W., Cai, Y., Feng, Y.: Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity. SIAM J. Numer. Anal. 60, 1962\u20131984 (2022)","journal-title":"SIAM J. Numer. Anal."},{"key":"1745_CR26","unstructured":"Feng, Y., Schratz, K.: Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine-Gordon equation. arXiv:2211.09402"},{"key":"1745_CR27","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1007\/s10915-022-02061-w","volume":"94","author":"X Yin","year":"2023","unstructured":"Yin, X., Cao, W.: A class of efficient Hamiltonian conservative spectral methods for Korteweg-de Vries equations. J. Sci. Comput. 94, 10 (2023)","journal-title":"J. Sci. Comput."},{"key":"1745_CR28","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1090\/mcom\/3694","volume":"91","author":"W Bao","year":"2022","unstructured":"Bao, W., Feng, Y., Su, C.: Uniform error bounds of time-splitting spectral methods for the long-time dynamics of the nonlinear Klein-Gordon equation with weak nonlinearity. Math. Comp. 91, 811\u2013842 (2022)","journal-title":"Math. Comp."},{"key":"1745_CR29","doi-asserted-by":"crossref","first-page":"897","DOI":"10.1002\/num.22558","volume":"37","author":"Y Feng","year":"2021","unstructured":"Feng, Y.: Long time error analysis of the fourth-order compact finite difference methods for the nonlinear Klein-Gordon equation with weak nonlinearity. Numer. Methods Partial Differential Equations 37, 897\u2013914 (2021)","journal-title":"Numer. Methods Partial Differential Equations"},{"key":"1745_CR30","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1007\/s10915-021-01558-0","volume":"88","author":"Y Feng","year":"2021","unstructured":"Feng, Y.: Improved error bounds of the Strang splitting method for the highly oscillatory fractional nonlinear Schr\u00f6dinger equation. J. Sci. Comput. 88, 48 (2021)","journal-title":"J. Sci. Comput."},{"key":"1745_CR31","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1007\/s11075-012-9689-0","volume":"64","author":"J Huang","year":"2013","unstructured":"Huang, J., Tang, Y., V\u00e1zquez, L., Yang, J.: Two finite difference schemes for time fractional diffusion-wave equation. Numer. Algorithms 64, 707\u2013720 (2013)","journal-title":"Numer. Algorithms"},{"key":"1745_CR32","volume":"21","author":"J Jia","year":"2021","unstructured":"Jia, J., Xu, H., Jiang, X.: Fast evaluation for the two-dimensional nonlinear coupled time-space fractional Klein-Gordon-Zakharov equations. Appl. Math. Lett. 21, 107148 (2021)","journal-title":"Appl. Math. Lett."},{"key":"1745_CR33","doi-asserted-by":"crossref","first-page":"1581","DOI":"10.1007\/s11075-022-01480-y","volume":"93","author":"Y Wang","year":"2023","unstructured":"Wang, Y., Zhao, Y., Chen, H.: Discrete comparison principle of a finite difference method for the multi-term time fractional diffusion equation. Numer. Algorithms 93, 1581\u20131593 (2023)","journal-title":"Numer. Algorithms"},{"key":"1745_CR34","volume-title":"Fractional differential equations","author":"I Podlubny","year":"1999","unstructured":"Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)"},{"key":"1745_CR35","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","volume":"64","author":"H Sun","year":"2018","unstructured":"Sun, H., Zhang, Y., Baleanu, D., Chen, W., Chen, Y.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213\u2013231 (2018)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"1745_CR36","doi-asserted-by":"crossref","first-page":"2599","DOI":"10.1137\/130934192","volume":"52","author":"F Zeng","year":"2014","unstructured":"Zeng, F., Liu, F., Li, C., Burrage, K., Turner, I., Anh, V.: A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM J. Numer. Anal. 52, 2599\u20132622 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"1745_CR37","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1007\/s11075-017-0385-y","volume":"78","author":"P Lyu","year":"2018","unstructured":"Lyu, P., Vong, S.: A linearized second-order scheme for nonlinear time fractional Klein-Gordon type equations. Numer. Algorithms 78, 485\u2013511 (2018)","journal-title":"Numer. Algorithms"},{"key":"1745_CR38","volume":"405","author":"H Zhang","year":"2019","unstructured":"Zhang, H., Jiang, X., Zeng, F., Karniadakis, G.E.: A stabilized semi-implicit Fourier spectral method for nonlinear space-fractional reaction-diffusion equations. J. Comput. Phys. 405, 109141 (2019)","journal-title":"J. Comput. Phys."},{"key":"1745_CR39","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.enganabound.2014.09.008","volume":"50","author":"M Dehghan","year":"2015","unstructured":"Dehghan, M., Abbaszadeh, M., Mohebbi, A.: An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations. Eng. Anal. Bound. Elem. 50, 412\u2013434 (2015)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"1745_CR40","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1016\/j.chaos.2005.08.199","volume":"28","author":"W Chen","year":"2006","unstructured":"Chen, W.: Time-space fabric underlying anomalous diffusion. Chaos, Solitons Fractals 28, 923\u2013929 (2006)","journal-title":"Chaos, Solitons Fractals"},{"key":"1745_CR41","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1016\/j.apnum.2023.03.003","volume":"188","author":"H Mohebalizadeh","year":"2023","unstructured":"Mohebalizadeh, H., Adibi, H., Dehghan, M.: On the fractional Laplacian of some positive definite kernels with applications in numerically solving the surface quasi-geostrophic equation as a prominent fractional calculus model. Appl. Numer. Math. 188, 75\u201387 (2023)","journal-title":"Appl. Numer. Math."},{"key":"1745_CR42","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1016\/j.apm.2022.04.009","volume":"108","author":"AS Hendy","year":"2022","unstructured":"Hendy, A.S., Taha, T.R., Suragan, D., Zaky, M.A.: An energy-preserving computational approach for the semilinear space fractional damped Klein-Gordon equation with a generalized scalar potential. Appl. Math. Model. 108, 512\u2013530 (2022)","journal-title":"Appl. Math. Model."},{"key":"1745_CR43","volume":"386","author":"B Hou","year":"2021","unstructured":"Hou, B., Liang, D.: Time fourth-order energy-preserving AVF finite difference method for nonlinear space-fractional wave equations. J. Comput. Appl. Math. 386, 113227 (2021)","journal-title":"J. Comput. Appl. Math."},{"key":"1745_CR44","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/j.matcom.2021.03.034","volume":"188","author":"D Hu","year":"2021","unstructured":"Hu, D., Cai, W., Xu, Z., Bo, Y., Wang, Y.: Dissipation-preserving Fourier pseudo-spectral method for the space fractional nonlinear sine-Gordon equation with damping. Math. Comput. Simul. 188, 35\u201359 (2021)","journal-title":"Math. Comput. Simul."},{"key":"1745_CR45","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.cnsns.2016.11.002","volume":"46","author":"JE Mac\u00edas-D\u00edaz","year":"2017","unstructured":"Mac\u00edas-D\u00edaz, J.E.: Numerical study of the process of nonlinear supratransmission in Riesz space-fractional sine-Gordon equations. Commun. Nonlinear Sci. Numer. Simul. 46, 89\u2013102 (2017)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"1745_CR46","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.jcp.2017.09.028","volume":"351","author":"JE Mac\u00edas-D\u00edaz","year":"2017","unstructured":"Mac\u00edas-D\u00edaz, J.E.: A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives. J. Comput. Phys. 351, 40\u201358 (2017)","journal-title":"J. Comput. Phys."},{"key":"1745_CR47","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.apnum.2020.09.009","volume":"159","author":"Z Xing","year":"2021","unstructured":"Xing, Z., Wen, L., Xiao, H.: A fourth-order conservative difference scheme for the Riesz space-fractional Sine-Gordon equations and its fast implementation. Appl. Numer. Math. 159, 221\u2013238 (2021)","journal-title":"Appl. Numer. Math."},{"key":"1745_CR48","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-30726-6","volume-title":"Spectral methods","author":"C Canuto","year":"2006","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods. Springer-Verlag, Germany (2006)"},{"key":"1745_CR49","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71041-7","volume-title":"Spectral methods: algorithms","author":"J Shen","year":"2011","unstructured":"Shen, J., Tang, T., Wang, L.: Spectral methods: algorithms. Analysis and Applications. Springer, Heidelberg (2011)"},{"key":"1745_CR50","doi-asserted-by":"crossref","first-page":"372","DOI":"10.1137\/0704033","volume":"4","author":"JD Lawson","year":"1967","unstructured":"Lawson, J.D.: Generalized Runge-Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4, 372\u2013380 (1967)","journal-title":"SIAM J. Numer. Anal."},{"key":"1745_CR51","doi-asserted-by":"crossref","first-page":"2399","DOI":"10.1093\/imanum\/drz030","volume":"40","author":"A Ostermann","year":"2020","unstructured":"Ostermann, A., Su, C.: A Lawson-type exponential integrator for the Korteweg-de Vries equation. IMA J. Numer. Anal. 40, 2399\u20132414 (2020)","journal-title":"IMA J. Numer. Anal."},{"key":"1745_CR52","first-page":"468","volume":"204","author":"B Guo","year":"2008","unstructured":"Guo, B., Han, Y., Xin, J.: Existence of the global smooth solution to the period boundary value problem of fractional nonlinear Schr\u00f6dinger equation. Appl. Math. Comput. 204, 468\u2013477 (2008)","journal-title":"Appl. Math. Comput."},{"key":"1745_CR53","doi-asserted-by":"crossref","first-page":"1629","DOI":"10.1007\/s10915-019-01001-5","volume":"80","author":"C Jiang","year":"2019","unstructured":"Jiang, C., Cai, W., Wang, Y.: A linearly implicit and local energy-preserving scheme for the Sine-Gordon equation based on the invariant energy quadratization approach. J. Sci. Comput. 80, 1629\u20131655 (2019)","journal-title":"J. Sci. Comput."},{"key":"1745_CR54","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1016\/j.camwa.2021.03.009","volume":"90","author":"J Wang","year":"2021","unstructured":"Wang, J., Huang, Q.: A family of effective structure-preserving schemes with second-order accuracy for the undamped sine-Gordon equation. Comput. Math. Appl. 90, 38\u201345 (2021)","journal-title":"Comput. Math. Appl."},{"key":"1745_CR55","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1007\/BF00042761","volume":"29","author":"K Djidjeli","year":"1995","unstructured":"Djidjeli, K., Price, W.G., Twizell, E.H.: Numerical solutions of a damped sine-Gordon equation in two space variables. J. Engrg. Math. 29, 347\u2013369 (1995)","journal-title":"J. Engrg. Math."},{"key":"1745_CR56","volume":"119","author":"T Ma","year":"2023","unstructured":"Ma, T., Zheng, Q., Fu, Y.: Optimal error estimation of two fast structure-preserving algorithms for the Riesz fractional sine-Gordon equation. Commun. Nonlinear Sci. Numer. Simul. 119, 107067 (2023)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01745-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01745-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01745-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T10:12:21Z","timestamp":1728295941000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01745-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,2,10]]},"references-count":56,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,11]]}},"alternative-id":["1745"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01745-0","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2024,2,10]]},"assertion":[{"value":"1 August 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 December 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 February 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}]}}