{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T12:27:52Z","timestamp":1774873672378,"version":"3.50.1"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2007,2,14]],"date-time":"2007-02-14T00:00:00Z","timestamp":1171411200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2007,2,28]]},"DOI":"10.1007\/s11075-006-9061-3","type":"journal-article","created":{"date-parts":[[2007,2,13]],"date-time":"2007-02-13T03:26:36Z","timestamp":1171337196000},"page":"295-308","source":"Crossref","is-referenced-by-count":51,"title":["A modified predictor\u2013corrector scheme for the two-dimensional sine-Gordon equation"],"prefix":"10.1007","volume":"43","author":[{"given":"A. G.","family":"Bratsos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2007,2,14]]},"reference":[{"key":"9061_CR1","doi-asserted-by":"crossref","unstructured":"Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. London Math. Soc. Lecture Note Ser. 149 (1991)","DOI":"10.1017\/CBO9780511623998"},{"key":"9061_CR2","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. Engrg. 86, 1\u201326 (1991)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"issue":"22","key":"9061_CR3","doi-asserted-by":"crossref","first-page":"16139","DOI":"10.1103\/PhysRevB.54.16139","volume":"54","author":"A. Benabdallah","year":"1996","unstructured":"Benabdallah, A., Caputo, J.G., Scott, A.C.: Exponentially tapered Josephson flux-flow oscillator. Phys. Rev. B 54(22), 16139\u201316146 (1996)","journal-title":"Phys. Rev. B"},{"issue":"1","key":"9061_CR4","first-page":"12","volume":"24","author":"I.L. Bogolyubski\u012d","year":"1976","unstructured":"Bogolyubski\u012d, I.L., Makhankov, V.G.: Lifetime of pulsating solitons in certain classical models. JETP Lett. 24(1), 12\u201314 (1976)","journal-title":"JETP Lett."},{"issue":"10","key":"9061_CR5","first-page":"535","volume":"24","author":"I.L. Bogolyubski\u012d","year":"1976","unstructured":"Bogolyubski\u012d, I.L.: Oscillating particle-like solutions of the nonlinear Klein\u2013Gordon equation. JETP Lett. 24(10), 535\u2013538 (1976)","journal-title":"JETP Lett."},{"issue":"2","key":"9061_CR6","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1002\/anac.200410035","volume":"2","author":"A.G. Bratsos","year":"2005","unstructured":"Bratsos, A.G.: An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions. Appl. Numer. Anal. Comput. Math. 2(2), 189\u2013211 (2005)","journal-title":"Appl. Numer. Anal. Comput. Math."},{"key":"9061_CR7","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1088\/0031-8949\/55\/2\/002","volume":"55","author":"P.L. Christiansen","year":"1997","unstructured":"Christiansen, P.L., Gr\u00f8nbech-Jensen, N., Lomdahl, P.S., Malomed, B.A.: Oscillations of eccentric Pulsons. Phys. Scripta 55, 131\u2013134 (1997)","journal-title":"Phys. Scripta"},{"issue":"2","key":"9061_CR8","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/0375-9601(78)90798-3","volume":"68A","author":"P.L. Christiansen","year":"1978","unstructured":"Christiansen, P.L., Olsen, O.H.: Return effect for rotationally symmetric solitary wave solutions to the sine-Gordon equation. Phys. Lett. 68A(2), 185\u2013188 (1978)","journal-title":"Phys. Lett."},{"key":"9061_CR9","first-page":"30","volume":"59","author":"P.L. Christiansen","year":"1979","unstructured":"Christiansen, P.L., Olsen, O.H.: On dynamical two-dimensional solutions to the sine-Gordon equation. Z. Angew. Math. Mech. 59, 30 (1979)","journal-title":"Z. Angew. Math. Mech."},{"key":"9061_CR10","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1088\/0031-8949\/20\/3-4\/032","volume":"20","author":"P.L. Christiansen","year":"1979","unstructured":"Christiansen, P.L., Olsen, O.H.: Ring-shaped quasi-soliton solutions to the two and three-dimensional sine-Gordon equations. Phys. Scripta 20, 531\u2013538 (1979)","journal-title":"Phys. Scripta"},{"key":"9061_CR11","first-page":"482","volume":"2D","author":"P.L. Christiansen","year":"1981","unstructured":"Christiansen, P.L., Lomdahl, P.S.: Numerical solutions of 2+1 dimensional sine-Gordon solitons. Physica 2D, 482\u2013494 (1981)","journal-title":"Physica"},{"key":"9061_CR12","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."},{"issue":"1","key":"9061_CR13","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1103\/PhysRevE.62.R53","volume":"62","author":"Y.B. Gaididei","year":"2000","unstructured":"Gaididei, Y.B., Mingaleev, S.F., Christiansen, P.L.: Curvature-induced symmetry breaking in nonlinear Schr\u00f6dinger models. Phys. Rev. E 62(1), 53\u201356 (2000)","journal-title":"Phys. Rev. E"},{"issue":"4","key":"9061_CR14","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0375-9601(83)90570-4","volume":"98A","author":"J. Geicke","year":"1983","unstructured":"Geicke, J.: On the reflection of radially symmetrical sine-Gordon kinks in the origin. Phys. Lett. 98A(4), 147\u2013150 (1983)","journal-title":"Phys. Lett."},{"key":"9061_CR15","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1088\/0031-8949\/29\/5\/003","volume":"29","author":"J. Geicke","year":"1984","unstructured":"Geicke, J.: Cylindrical pulsons in nonlinear relativistic wave equations. Phys. Scripta 29, 431\u2013434 (1984)","journal-title":"Phys. Scripta"},{"key":"9061_CR16","first-page":"1","volume":"25B","author":"J.D. Gibbon","year":"1976","unstructured":"Gibbon, J.D., Zambotti, G.: The interaction of n-dimensional soliton wave fronts. Nuovo Cimento 25B, 1 (1976)","journal-title":"Nuovo Cimento"},{"key":"9061_CR17","first-page":"1","volume":"69","author":"C. Gorria","year":"2004","unstructured":"Gorria, C., Gaididei, Y. B., Soerensen, M.P., Christiansen, P.L., Caputo, J.G.: Kink propagation and trapping in a two-dimensional curved Josephson junction. Phys. Rev. B 69, 1\u201310 (2004)","journal-title":"Phys. Rev. B"},{"key":"9061_CR18","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1007\/BF02770537","volume":"23","author":"G. Grella","year":"1978","unstructured":"Grella, G., Marinaro, M.: Special solution of the sine-Gordon equation in 2+1 dimensions. Lett. Nuovo Cimento 23, 459 (1978)","journal-title":"Lett. Nuovo Cimento"},{"key":"9061_CR19","first-page":"1","volume":"18","author":"B.-Y. Guo","year":"1986","unstructured":"Guo, B.-Y., Pascual, P.J., Rodriguez, M.J., V\u00e1zquez, L.: Numerical solution of the sine-Gordon equation. Appl. Math. Comput. 18, 1\u201314 (1986)","journal-title":"Appl. Math. Comput."},{"issue":"19","key":"9061_CR20","doi-asserted-by":"crossref","first-page":"3803","DOI":"10.1103\/PhysRevLett.82.3803","volume":"82","author":"T. Harayama","year":"1999","unstructured":"Harayama, T., Davis, P., Ikeda, K.S.: Nonlinear whispering gallery modes. Phys. Rev. Lett. 82(19), 3803\u20133806 (1999)","journal-title":"Phys. Rev. Lett."},{"key":"9061_CR21","doi-asserted-by":"crossref","first-page":"1566","DOI":"10.1143\/JPSJ.35.1566","volume":"35","author":"R. Hirota","year":"1973","unstructured":"Hirota, R.: Exact three-soliton solution of the two-dimensional sine-Gordon equation. J. Phys. Soc. Japan 35, 1566 (1973)","journal-title":"J. Phys. Soc. Japan"},{"key":"9061_CR22","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1088\/0305-4470\/12\/10\/002","volume":"12","author":"P. Kaliappan","year":"1979","unstructured":"Kaliappan, P., Lakshmanan, M.: Kadomtsev-Petviashvili and two-dimensional sine-Gordon equations: reduction to Painlev\u00e8 transcendents. J. Phys. A: Math. Gen. 12, 249 (1979)","journal-title":"J. Phys. A: Math. Gen."},{"key":"9061_CR23","doi-asserted-by":"crossref","first-page":"1091","DOI":"10.1143\/JPSJ.41.1091","volume":"41","author":"K.K. Kobayashi","year":"1976","unstructured":"Kobayashi, K.K., Izutsu, M.: Exact solution of the n-dimensional sine-Gordon equation. J. Phys. Soc. Japan. 41, 1091 (1976)","journal-title":"J. Phys. Soc. Japan"},{"issue":"4","key":"9061_CR24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1103\/PhysRevLett.89.040401","volume":"89","author":"A.E. Leanhardt","year":"2002","unstructured":"Leanhardt, A.E., Chikkatur, A.P., Kielpinski, D., Shin, Y., Gustavson, T.L., Ketterle, W., Pritchard, D.E.: Propagation of Bose\u2013Einstein condensates in a magnetic waveguide. Phys. Rev. Lett. 89(4), 1\u20134\/040401 (2002)","journal-title":"Phys. Rev. Lett."},{"key":"9061_CR25","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1103\/PhysRevLett.41.435","volume":"41","author":"G. Leibbrandt","year":"1978","unstructured":"Leibbrandt, G.: New exact solutions of the classical sine-Gordon equation in 2+1 and 3+1 dimensions. Phys. Rev. Lett. 41, 435\u2013438 (1978)","journal-title":"Phys. Rev. Lett."},{"key":"9061_CR26","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/0167-2789(87)90071-6","volume":"24","author":"B.A. Malomed","year":"1987","unstructured":"Malomed, B.A.: Decay of shrinking solitons in multidimensional sine-Gordon equation. Phys. D 24, 155\u2013171 (1987)","journal-title":"Phys. D"},{"key":"9061_CR27","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/0167-2789(91)90118-S","volume":"52","author":"B.A. Malomed","year":"1991","unstructured":"Malomed, B.A.: Dynamic of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model. Phys. D 52, 157\u2013170 (1991)","journal-title":"Phys. D"},{"key":"9061_CR28","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1016\/S0167-2789(85)80009-9","volume":"15","author":"E.M. Maslov","year":"1985","unstructured":"Maslov, E.M.: Dynamics of rotationally symmetric solitons in near-SG field model with applications to large-area Josephson junctions and ferromagnets. Phys. D 15, 433\u2013443 (1985)","journal-title":"Phys. D"},{"issue":"6","key":"9061_CR29","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/0375-9601(88)90789-X","volume":"131","author":"E.M. Maslov","year":"1988","unstructured":"Maslov, E.M.: Rotationally symmetric sG oscillator with tunable frequency. Phys. Lett. A 131(6), 364\u2013367 (1988)","journal-title":"Phys. Lett. A"},{"key":"9061_CR30","doi-asserted-by":"crossref","first-page":"5777","DOI":"10.1103\/PhysRevE.62.5777","volume":"62","author":"S.F. Mingaleev","year":"2000","unstructured":"Mingaleev, S.F., Kivshar, Y.S., Sammut, R.A.: Long-range interaction and nonlinear localized modes in photonic crystal waveguides. Phys. Rev. E 62, 5777\u20135782 (2000)","journal-title":"Phys. Rev. E"},{"issue":"9","key":"9061_CR31","doi-asserted-by":"crossref","first-page":"4095","DOI":"10.1063\/1.1663917","volume":"45","author":"K. Nakajima","year":"1974","unstructured":"Nakajima, K., Onodera, Y., Nakamura, T., Sato, R.: Numerical analysis of vortex motion on Josephson structures. J. Appl. Phys. 45(9), 4095\u20134099 (1974)","journal-title":"J. Appl. Phys."},{"key":"9061_CR32","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. Simulation 68, 355\u2013373 (2005)","journal-title":"Math. Comput. Simulation"},{"issue":"1","key":"9061_CR33","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1103\/PhysRevLett.84.151","volume":"84","author":"A. Wallraff","year":"2000","unstructured":"Wallraff, A., Ustinov, A.V., Kurin, V.V., Shereshevsky, I.A., Vdovicheva, N.K.: Whispering vortices. Phys. Rev. Lett. 84(1), 151\u2013154 (2000)","journal-title":"Phys. Rev. Lett."},{"key":"9061_CR34","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1016\/S0167-2789(99)00128-1","volume":"135","author":"J.X. Xin","year":"2000","unstructured":"Xin, J.X.: Modeling light bullets with the two-dimensional sine-Gordon equation. Phys. D 135, 345\u2013368 (2000)","journal-title":"Phys. D"},{"key":"9061_CR35","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1016\/0375-9601(79)90469-9","volume":"72A","author":"J. Zagrodzinsky","year":"1979","unstructured":"Zagrodzinsky, J.: Particular solutions of the sine-Gordon equation in 2+1 dimensions. Phys. Lett. 72A, 284\u2013286 (1979)","journal-title":"Phys. Lett."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-006-9061-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-006-9061-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-006-9061-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T01:08:08Z","timestamp":1559351288000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-006-9061-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,2,14]]},"references-count":35,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2007,2,28]]}},"alternative-id":["9061"],"URL":"https:\/\/doi.org\/10.1007\/s11075-006-9061-3","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,2,14]]}}}