{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T20:09:40Z","timestamp":1773518980164,"version":"3.50.1"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2019,4,16]],"date-time":"2019-04-16T00:00:00Z","timestamp":1555372800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,4,16]],"date-time":"2019-04-16T00:00:00Z","timestamp":1555372800000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1615718"],"award-info":[{"award-number":["1615718"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"AMS-Simons Travel Grant Program"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2019,10]]},"DOI":"10.1007\/s00332-019-09542-7","type":"journal-article","created":{"date-parts":[[2019,4,16]],"date-time":"2019-04-16T12:03:10Z","timestamp":1555416190000},"page":"2185-2229","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":74,"title":["Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schr\u00f6dinger Equation"],"prefix":"10.1007","volume":"29","author":[{"given":"Deniz","family":"Bilman","sequence":"first","affiliation":[]},{"given":"Robert","family":"Buckingham","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,4,16]]},"reference":[{"key":"9542_CR1","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1016\/S0375-9601(00)00020-7","volume":"267","author":"M Ablowitz","year":"2000","unstructured":"Ablowitz, M., Chakravarty, S., Trubatch, A., Villarroel, J.: A novel class of solutions of the non-stationary Schr\u00f6dinger and the Kadomtsev\u2013Petviashvili I equations. Phys. Lett. A 267, 132\u2013146 (2000)","journal-title":"Phys. Lett. A"},{"key":"9542_CR2","doi-asserted-by":"publisher","first-page":"2171","DOI":"10.1088\/0266-5611\/23\/5\/021","volume":"23","author":"T Aktosun","year":"2007","unstructured":"Aktosun, T., Demontis, F., van der Mee, C.: Exact solutions to the focusing nonlinear Schr\u00f6dinger equation. Inverse Problems 23, 2171\u20132195 (2007)","journal-title":"Inverse Problems"},{"key":"9542_CR3","doi-asserted-by":"publisher","first-page":"123521","DOI":"10.1063\/1.3520596","volume":"51","author":"T Aktosun","year":"2010","unstructured":"Aktosun, T., Demontis, F., van der Mee, C.: Exact solutions to the sine-Gordon equation. J. Math. Phys. 51, 123521 (2010)","journal-title":"J. Math. Phys."},{"key":"9542_CR4","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1002\/cpa.3160370105","volume":"37","author":"R Beals","year":"1984","unstructured":"Beals, R., Coifman, R.: Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math. 37, 39\u201390 (1984)","journal-title":"Commun. Pure Appl. Math."},{"key":"9542_CR5","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1002\/sapm1967461133","volume":"46","author":"D Benney","year":"1967","unstructured":"Benney, D., Newell, A.: The propagation of nonlinear wave envelopes. J. Math. Phys. 46, 133\u2013139 (1967)","journal-title":"J. Math. Phys."},{"key":"9542_CR6","doi-asserted-by":"publisher","first-page":"678","DOI":"10.1002\/cpa.21445","volume":"66","author":"M Bertola","year":"2013","unstructured":"Bertola, M., Tovbis, A.: Universality for the focusing nonlinear Schr\u00f6dinger equation at the gradient catastrophe point: rational breathers and poles of the tritronqu\u00e9e solution to Painlev\u00e9 I. Commun. Pure Appl. Math. 66, 678\u2013752 (2013)","journal-title":"Commun. Pure Appl. Math."},{"key":"9542_CR7","unstructured":"Bilman, D., Ling, L., Miller, P.: Extreme superposition: rogue waves of infinite order and the Painlev\u00e9-III hierarchy. \n                    arXiv:1806.00545\n                    \n                   (2018)"},{"key":"9542_CR8","doi-asserted-by":"publisher","unstructured":"Bilman, D., Miller, P.: A robust inverse scattering transform for the focusing nonlinear Schr\u00f6dinger equation. Commun. Pure Appl. Math. (2019). \n                    https:\/\/doi.org\/10.1002\/cpa.21819","DOI":"10.1002\/cpa.21819"},{"key":"9542_CR9","doi-asserted-by":"publisher","first-page":"1015","DOI":"10.1007\/s00220-017-2897-7","volume":"354","author":"R Buckingham","year":"2017","unstructured":"Buckingham, R., Jenkins, R., Miller, P.: Semiclassical soliton ensembles for the three-wave resonant interaction equations. Commun. Math. Phys. 354, 1015\u20131100 (2017)","journal-title":"Commun. Math. Phys."},{"key":"9542_CR10","doi-asserted-by":"publisher","first-page":"2296","DOI":"10.1016\/j.physd.2008.02.010","volume":"237","author":"R Buckingham","year":"2008","unstructured":"Buckingham, R., Miller, P.: Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation. Physica D 237, 2296\u20132341 (2008)","journal-title":"Physica D"},{"key":"9542_CR11","doi-asserted-by":"publisher","first-page":"2489","DOI":"10.1088\/0951-7715\/27\/10\/2489","volume":"27","author":"R Buckingham","year":"2014","unstructured":"Buckingham, R., Miller, P.: Large-degree asymptotics of rational Painlev\u00e9-II functions: noncritial behaviour. Nonlinearity 27, 2489\u20132577 (2014)","journal-title":"Nonlinearity"},{"key":"9542_CR12","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1103\/PhysRevLett.13.479","volume":"13","author":"R Chiao","year":"1964","unstructured":"Chiao, R., Garmire, E., Townes, C.: Self-trapping of optical beams. Phys. Rev. Lett. 13, 479\u2013482 (1964)","journal-title":"Phys. Rev. Lett."},{"key":"9542_CR13","doi-asserted-by":"publisher","first-page":"295","DOI":"10.2307\/2946540","volume":"137","author":"P Deift","year":"1993","unstructured":"Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann\u2013Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. (2) 137, 295\u2013368 (1993)","journal-title":"Ann. Math. (2)"},{"key":"9542_CR14","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/j.physd.2016.04.006","volume":"333","author":"G El","year":"2016","unstructured":"El, G., Hoefer, M.: Dispersive shock waves and modulation theory. Physica D 333, 11\u201365 (2016)","journal-title":"Physica D"},{"key":"9542_CR15","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1088\/0305-4470\/20\/2\/023","volume":"20","author":"B Fuchssteiner","year":"1987","unstructured":"Fuchssteiner, B., Aiyer, R.: Multisolitons, or the discrete eigenfunctions of the recursion operator of non-linear evolution equations: II. Background. J. Phys. A 20, 375\u2013388 (1987)","journal-title":"J. Phys. A"},{"key":"9542_CR16","doi-asserted-by":"publisher","first-page":"619","DOI":"10.1364\/OL.19.000619","volume":"19","author":"L Gagnon","year":"1994","unstructured":"Gagnon, L., Sti\u00e9venart, N.: $$N$$-soliton interaction in optical fibers: the multiple-pole case. Opt. Lett. 19, 619\u2013621 (1994)","journal-title":"Opt. Lett."},{"key":"9542_CR17","volume-title":"Semiclassical Soliton Ensembles for the Focusing Nonlinear Schr\u00f6dinger Equation, Annals of Mathematics Studies","author":"S Kamvissis","year":"2003","unstructured":"Kamvissis, S., McLaughlin, K., Miller, P.: Semiclassical Soliton Ensembles for the Focusing Nonlinear Schr\u00f6dinger Equation, Annals of Mathematics Studies, vol. 154. Princeton University Press, Princeton (2003)"},{"key":"9542_CR18","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/j.aml.2016.11.008","volume":"66","author":"Y Kuang","year":"2017","unstructured":"Kuang, Y., Zhu, J.: The higher-order soliton solutions for the coupled Sasa-Satsuma system via the $${\\overline{\\partial }}$$-dressing method. Appl. Math. Lett. 66, 47\u201353 (2017)","journal-title":"Appl. Math. Lett."},{"key":"9542_CR19","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1016\/j.physd.2016.03.012","volume":"327","author":"L Ling","year":"2016","unstructured":"Ling, L., Feng, B., Zhu, Z.: Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation. Physica D 327, 13\u201329 (2016)","journal-title":"Physica D"},{"key":"9542_CR20","doi-asserted-by":"publisher","first-page":"951","DOI":"10.1002\/cpa.20162","volume":"60","author":"G Lyng","year":"2007","unstructured":"Lyng, G., Miller, P.: The $$N$$-soliton of the focusing nonlinear Schr\u00f6dinger equation for $$N$$ large. Commun. Pure Appl. Math. 60, 951\u20131026 (2007)","journal-title":"Commun. Pure Appl. Math."},{"key":"9542_CR21","first-page":"1","volume":"179","author":"T Martines","year":"2017","unstructured":"Martines, T.: Generalized inverse scattering transform for the nonlinear Schr\u00f6dinger equation for bound states with higher multiplicities. Electron. J. Differ. Equ. 179, 1\u201315 (2017)","journal-title":"Electron. J. Differ. Equ."},{"key":"9542_CR22","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-00922-2","volume-title":"Darboux Transformations and Solitons","author":"V Matveev","year":"1991","unstructured":"Matveev, V., Salle, M.: Darboux Transformations and Solitons. Springer, Berlin (1991)"},{"key":"9542_CR23","doi-asserted-by":"publisher","first-page":"330","DOI":"10.1016\/0167-2789(87)90107-2","volume":"25","author":"E Olmedilla","year":"1987","unstructured":"Olmedilla, E.: Multiple pole solutions of the non-linear Schr\u00f6dinger equation. Physica D 25, 330\u2013346 (1987)","journal-title":"Physica D"},{"key":"9542_CR24","unstructured":"Olver, S.: RHPackage (2011). \n                    http:\/\/www.maths.usyd.edu.au\/u\/olver\/projects\/RHPackage.html"},{"key":"9542_CR25","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/0167-2789(83)90295-6","volume":"9","author":"C P\u00f6ppe","year":"1983","unstructured":"P\u00f6ppe, C.: Construction of solutions of the sine-Gordon equation by means of Fredholm determinants. Physica D 9, 103\u2013139 (1983)","journal-title":"Physica D"},{"key":"9542_CR26","doi-asserted-by":"publisher","first-page":"025210","DOI":"10.1088\/1751-8113\/42\/2\/025210","volume":"42","author":"A Sakka","year":"2009","unstructured":"Sakka, A.: Linear problems and hierarchies of Painlev\u00e9 equations. J. Phys. A 42, 025210 (2009)","journal-title":"J. Phys. A"},{"key":"9542_CR27","doi-asserted-by":"publisher","first-page":"2930","DOI":"10.1088\/1361-6544\/aa6d9a","volume":"30","author":"C Schiebold","year":"2017","unstructured":"Schiebold, C.: Asymptotics for the multiple pole solutions of the nonlinear Schr\u00f6dinger equation. Nonlinearity 30, 2930\u20132981 (2017)","journal-title":"Nonlinearity"},{"key":"9542_CR28","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1111\/1467-9590.00240","volume":"110","author":"V Shchesnovich","year":"2003","unstructured":"Shchesnovich, V., Yang, J.: Higher-order solitons in the $$N$$-wave system. Stud. Appl. Math. 110, 297\u2013332 (2003)","journal-title":"Stud. Appl. Math."},{"key":"9542_CR29","doi-asserted-by":"publisher","first-page":"400","DOI":"10.1134\/S0021364017180126","volume":"106","author":"B Suleimanov","year":"2017","unstructured":"Suleimanov, B.: Effect of a small dispersion on self-focusing in a spatially one-dimensional case. JETP Lett. 106, 400\u2013405 (2017)","journal-title":"JETP Lett."},{"key":"9542_CR30","doi-asserted-by":"publisher","first-page":"329","DOI":"10.2977\/prims\/1195191998","volume":"10","author":"S Tanaka","year":"1975","unstructured":"Tanaka, S.: Non-linear Schr\u00f6dinger equation and modified Korteweg\u2013de Vries equation; construction of solutions in terms of scattering data. Publ. RIMS, Kyoto Univ. 10, 329\u2013357 (1975)","journal-title":"Publ. RIMS, Kyoto Univ."},{"key":"9542_CR31","volume-title":"Riemann\u2013Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions","author":"T Trogdon","year":"2016","unstructured":"Trogdon, T., Olver, S.: Riemann\u2013Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions. SIAM, Philadelphia (2016)"},{"key":"9542_CR32","doi-asserted-by":"publisher","first-page":"2908","DOI":"10.1143\/JPSJ.53.2908","volume":"53","author":"H Tsuru","year":"1984","unstructured":"Tsuru, H., Wadati, M.: The multiple pole solutions of the sine-Gordon equation. J. Phys. Soc. Jpn. 53, 2908\u20132921 (1984)","journal-title":"J. Phys. Soc. Jpn."},{"key":"9542_CR33","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s002200050716","volume":"207","author":"J Villarroel","year":"1999","unstructured":"Villarroel, J., Ablowitz, M.: On the discrete spectrum of the nonstationary Schr\u00f6dinger equation and multipole lumps of the Kadomtsev\u2013Petviashvili I equation. Commun. Math. Phys. 207, 1\u201342 (1999)","journal-title":"Commun. Math. Phys."},{"key":"9542_CR34","doi-asserted-by":"publisher","first-page":"4614","DOI":"10.1088\/1361-6544\/aa8cab","volume":"30","author":"N Vinh","year":"2017","unstructured":"Vinh, N.: Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation. Nonlinearity 30, 4614\u20134648 (2017)","journal-title":"Nonlinearity"},{"key":"9542_CR35","doi-asserted-by":"publisher","first-page":"2029","DOI":"10.1143\/JPSJ.51.2029","volume":"51","author":"M Wadati","year":"1982","unstructured":"Wadati, M., Ohkuma, K.: Multiple-pole solutions of the modified Korteweg\u2013de Vries equation. J. Phys. Soc. Jpn. 51, 2029\u20132035 (1982)","journal-title":"J. Phys. Soc. Jpn."},{"key":"9542_CR36","doi-asserted-by":"publisher","first-page":"190","DOI":"10.1007\/BF00913182","volume":"9","author":"V Zakharov","year":"1968","unstructured":"Zakharov, V.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190\u2013194 (1968)","journal-title":"J. Appl. Mech. Tech. Phys."},{"key":"9542_CR37","unstructured":"Zakharov, V., Shabat, A.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Physics JETP 34, 62\u201369 (1972). Translated from Z. Eksper. Teoret. Fiz. 61, 118\u2013134 (1971)"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-019-09542-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00332-019-09542-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-019-09542-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,5,19]],"date-time":"2020-05-19T09:53:08Z","timestamp":1589881988000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-019-09542-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,16]]},"references-count":37,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2019,10]]}},"alternative-id":["9542"],"URL":"https:\/\/doi.org\/10.1007\/s00332-019-09542-7","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,4,16]]},"assertion":[{"value":"26 July 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 April 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}