{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T06:28:04Z","timestamp":1780986484794,"version":"3.54.1"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2019,7,4]],"date-time":"2019-07-04T00:00:00Z","timestamp":1562198400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,7,4]],"date-time":"2019-07-04T00:00:00Z","timestamp":1562198400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2019,12]]},"DOI":"10.1007\/s00332-019-09559-y","type":"journal-article","created":{"date-parts":[[2019,7,4]],"date-time":"2019-07-04T16:03:45Z","timestamp":1562256225000},"page":"2797-2843","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":88,"title":["Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background"],"prefix":"10.1007","volume":"29","author":[{"given":"Jinbing","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5812-440X","authenticated-orcid":false,"given":"Dmitry E.","family":"Pelinovsky","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2019,7,4]]},"reference":[{"key":"9559_CR1","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1002\/sapm1974534249","volume":"53","author":"MJ Ablowitz","year":"1974","unstructured":"Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform: fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249\u2013315 (1974)","journal-title":"Stud. Appl. Math."},{"key":"9559_CR2","doi-asserted-by":"publisher","first-page":"809","DOI":"10.1007\/BF01017105","volume":"72","author":"NN Akhmediev","year":"1987","unstructured":"Akhmediev, N.N., Eleonskii, V.M., Kulagin, N.E.: Exact first-order solutions of the nonlinear Schr\u00f3dinger equation. Theor. Math. Phys. 72, 809\u2013818 (1987)","journal-title":"Theor. Math. Phys."},{"key":"9559_CR3","doi-asserted-by":"publisher","first-page":"3551","DOI":"10.1088\/0951-7715\/29\/11\/3551","volume":"29","author":"DS Agafontsev","year":"2016","unstructured":"Agafontsev, D.S., Zakharov, V.E.: Integrable turbulence generated from modulational instability of cnoidal waves. Nonlinearity 29, 3551\u20133578 (2016)","journal-title":"Nonlinearity"},{"key":"9559_CR4","volume-title":"Algebro-Geomtric Approach to Nonlinear Integrable Equations","author":"ED Belokolos","year":"1994","unstructured":"Belokolos, E.D., Bobenko, A.I., Enol\u015bkii, V.Z., Its, A.R., Matveev, V.B.: Algebro-Geomtric Approach to Nonlinear Integrable Equations. Springer, Berlin (1994)"},{"key":"9559_CR5","doi-asserted-by":"publisher","first-page":"20160340","DOI":"10.1098\/rspa.2016.0340","volume":"472","author":"M Bertola","year":"2016","unstructured":"Bertola, M., El, G.A., Tovbis, A.: Rogue waves in multiphase solutions of the focusing nonlinear Schr\u00f6dinger equation. Proc. R. Soc. Lond. A 472, 20160340 (2016)","journal-title":"Proc. R. Soc. Lond. A"},{"key":"9559_CR6","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1007\/s00220-017-2895-9","volume":"354","author":"M Bertola","year":"2017","unstructured":"Bertola, M., Tovbis, A.: Maximal amplitudes of finite-gap solutions for the focusing nonlinear Schr\u00f6dinger equation. Commun. Math. Phys. 354, 525\u2013547 (2017)","journal-title":"Commun. Math. Phys."},{"key":"9559_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/s00332-019-09542-7","author":"D Bilman","year":"2019","unstructured":"Bilman, D., Buckingham, R.: Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schr\u00f6dinger equation. J. Nonlinear Sci. (2019). https:\/\/doi.org\/10.1007\/s00332-019-09542-7","journal-title":"J. Nonlinear Sci."},{"key":"9559_CR8","doi-asserted-by":"publisher","first-page":"1722","DOI":"10.1002\/cpa.21819","volume":"72","author":"D Bilman","year":"2019","unstructured":"Bilman, D., Miller, P.D.: A robust inverse scattering transform for the focusing nonlinear Schr\u00f6dinger equation. Commun. Pure Appl. Math. 72, 1722\u20131805 (2019)","journal-title":"Commun. Pure Appl. Math."},{"key":"9559_CR9","doi-asserted-by":"publisher","first-page":"1141","DOI":"10.1017\/S0308210510001216","volume":"141","author":"JC Bronski","year":"2011","unstructured":"Bronski, J.C., Johnson, M.A., Kapitula, T.: An index theorem for the stability of periodic travelling waves of Korteweg-de Vries type. Proc. R. Soc. Edinburgh Sect. A 141, 1141\u20131173 (2011)","journal-title":"Proc. R. Soc. Edinburgh Sect. A"},{"key":"9559_CR10","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/978-3-319-20690-5_4","volume-title":"New Approaches to Nonlinear Waves","author":"Jared C. Bronski","year":"2016","unstructured":"Bronski, J.C., Hur, V.M., Johnson, M.A.: Modulational instability in equations of KdV type. In: New approaches to nonlinear waves, Lecture Notes in Phys, vol. 908, pp. 83\u2013133. Springer, Cham (2016)"},{"key":"9559_CR11","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1016\/j.wavemoti.2016.06.007","volume":"71","author":"A Calini","year":"2017","unstructured":"Calini, A., Schober, C.M.: Characterizing JONSWAP rogue waves and their statistics via inverse spectral data. Wave Motion 71, 5\u201317 (2017)","journal-title":"Wave Motion"},{"key":"9559_CR12","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1007\/978-3-642-84148-4_9","volume-title":"Nonlinear Physics","author":"Cewen Cao","year":"1990","unstructured":"Cao, C.W. and Geng, X.G.: Classical integrable systems generated through nonlinearization of eigenvalue problems. In: Nonlinear physics (Shanghai, 1989), pp. 68\u201378. Research Reports in Physics, Springer, Berlin (1990)"},{"key":"9559_CR13","doi-asserted-by":"publisher","first-page":"3948","DOI":"10.1063\/1.532936","volume":"40","author":"CW Cao","year":"1999","unstructured":"Cao, C.W., Wu, Y.T., Geng, X.G.: Relation between the Kadometsev-Petviashvili equation and the confocal involutive system. J. Math. Phys. 40, 3948\u20133970 (1999)","journal-title":"J. Math. Phys."},{"key":"9559_CR14","doi-asserted-by":"publisher","first-page":"1955","DOI":"10.1088\/1361-6544\/aaa2da","volume":"31","author":"J Chen","year":"2018","unstructured":"Chen, J., Pelinovsky, D.E.: Rogue periodic waves in the modified Korteweg-de Vries equation. Nonlinearity 31, 1955\u20131980 (2018)","journal-title":"Nonlinearity"},{"key":"9559_CR15","doi-asserted-by":"publisher","first-page":"104","DOI":"10.1140\/epjd\/e2016-70033-9","volume":"70","author":"A Chowdury","year":"2016","unstructured":"Chowdury, A., Ankiewicz, A., Akhmediev, N.: Periodic and rational solutions of modified Korteweg-de Vries equation. Eur. Phys. J. D 70, 104 (2016)","journal-title":"Eur. Phys. J. D"},{"key":"9559_CR16","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1070\/RM1981v036n02ABEH002596","volume":"36","author":"BA Dubrovin","year":"1981","unstructured":"Dubrovin, B.A.: Theta functions and nonlinear equations. Russ. Math. Surv. 36, 11\u201392 (1981)","journal-title":"Russ. Math. Surv."},{"key":"9559_CR17","doi-asserted-by":"publisher","first-page":"1433","DOI":"10.1088\/0951-7715\/14\/6\/302","volume":"14","author":"XG Geng","year":"2001","unstructured":"Geng, X.G., Cao, C.W.: Decomposition of the $$(2+1)$$-dimensional Gardner equation and its quasi-periodic solutions. Nonlinearity 14, 1433\u20131452 (2001)","journal-title":"Nonlinearity"},{"key":"9559_CR18","doi-asserted-by":"crossref","unstructured":"Gesztesy, F. and Holden, H.: Soliton Equations and Their Algebro-Geometric Solutions, volume I: $$(1+1)$$-Dimensional Continuous Models. Cambridge studies in advanced mathematics, vol. 79. Cambridge University Press, Cambridge (2003)","DOI":"10.1017\/CBO9780511546723"},{"issue":"563","key":"9559_CR19","first-page":"1","volume":"118","author":"F Gesztesy","year":"1995","unstructured":"Gesztesy, F., Svirsky, R.: (m)KdV solitons on the background of quasi-periodic finite-gap solutions. Mem. AMS 118(563), 1\u201388 (1995)","journal-title":"Mem. AMS"},{"key":"9559_CR20","volume-title":"Table of Integrals, Series and Products","author":"IS Gradshteyn","year":"2005","unstructured":"Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 6th edn. Academic Press, San Diego (2005)","edition":"6"},{"key":"9559_CR21","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1140\/epjst\/e2010-01249-4","volume":"185","author":"R Grimshaw","year":"2010","unstructured":"Grimshaw, R., Pelinovsky, E., Talipova, T., Sergeeva, A.: Rogue internal waves in the ocean: long wave model. Eur. Phys. J. Special Top. 185, 195\u2013208 (2010)","journal-title":"Eur. Phys. J. Special Top."},{"key":"9559_CR22","doi-asserted-by":"publisher","first-page":"5258","DOI":"10.1088\/1361-6544\/aaddcf","volume":"31","author":"PG Grinevich","year":"2018","unstructured":"Grinevich, P.G., Santini, P.M.: The finite gap method and the analytic description of the exact rogue wave recurrence in the periodic NLS Cauchy problem. Nonlinearity 31, 5258\u20135308 (2018)","journal-title":"Nonlinearity"},{"key":"9559_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/1-4020-3088-6","volume-title":"Darboux Transformation in Integrable Systems: Theory and their Applications to Geometry","author":"CH Gu","year":"2005","unstructured":"Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformation in Integrable Systems: Theory and their Applications to Geometry. Springer, Heidelberg (2005)"},{"key":"9559_CR24","doi-asserted-by":"publisher","first-page":"2945","DOI":"10.1088\/0305-4470\/23\/13\/031","volume":"23","author":"AM Kamchatnov","year":"1990","unstructured":"Kamchatnov, A.M.: On improving the effectiveness of periodic solutions of the NLS and DNLS equations. J. Phys. A: Math. Gen. 23, 2945\u20132960 (1990)","journal-title":"J. Phys. A: Math. Gen."},{"key":"9559_CR25","doi-asserted-by":"publisher","first-page":"036605","DOI":"10.1103\/PhysRevE.86.036605","volume":"86","author":"AM Kamchatnov","year":"2012","unstructured":"Kamchatnov, A.M., Kuo, Y.-H., Lin, T.-C., Horng, T.-L., Gou, S.-C., Clift, R., El, G.A., Grimshaw, R.H.J.: Undular bore theory for the Gardner equation. Phys. Rev. E 86, 036605 (2012)","journal-title":"Phys. Rev. E"},{"key":"9559_CR26","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1017\/jfm.2013.556","volume":"736","author":"AM Kamchatnov","year":"2013","unstructured":"Kamchatnov, A.M., Kuo, Y.-H., Lin, T.-C., Horng, T.-L., Gou, S.-C., Clift, R., El, G.A., Grimshaw, R.H.J.: Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation. J. Fluid Mech. 736, 495\u2013531 (2013)","journal-title":"J. Fluid Mech."},{"key":"9559_CR27","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1002\/cpa.3160460405","volume":"46","author":"C Kenig","year":"1993","unstructured":"Kenig, C., Ponce, G., Vega, L.: Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun. Pure Appl. Math. 46, 527\u2013620 (1993)","journal-title":"Commun. Pure Appl. Math."},{"key":"9559_CR28","volume-title":"Rogue Waves in the Ocean","author":"Ch Kharif","year":"2009","unstructured":"Kharif, Ch., Pelinovsky, E., Slunyaev, A.: Rogue Waves in the Ocean. Springer, Heidelberg (2009)"},{"key":"9559_CR29","doi-asserted-by":"crossref","unstructured":"Kibler, B., Chabchoub, A., Gelash, A., Akhmediev, N., Zakharov, V.E.: Ubiquitous nature of modulation instability: from periodic to localized perturbations. In: Wabnitz S. (ed.) Nonlinear Guided Wave Optics: A Testbed for Extreme Waves. Iop Publishing Ltd, Bristol (2018), chapter 7","DOI":"10.1088\/978-0-7503-1460-2ch7"},{"key":"9559_CR30","doi-asserted-by":"publisher","first-page":"3207","DOI":"10.1215\/00127094-2018-0033","volume":"167","author":"H Koch","year":"2018","unstructured":"Koch, H., Tataru, D.: Conserved energies for the cubic NLS in 1-d. Duke Math. J. 167, 3207\u20133313 (2018)","journal-title":"Duke Math. J."},{"key":"9559_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-00922-2","volume-title":"Darboux Transformations and Solitons","author":"VB Matveev","year":"1991","unstructured":"Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, New York (1991)"},{"key":"9559_CR32","first-page":"157","volume":"339","author":"MV Pavlov","year":"1994","unstructured":"Pavlov, M.V.: Double Lagrangian representation of KdV and general-solution of Whitham equations. Doklady Akad. Nauk 339, 157\u2013161 (1994)","journal-title":"Doklady Akad. Nauk"},{"key":"9559_CR33","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1134\/S1028335816090032","volume":"61","author":"EN Pelinovsky","year":"2016","unstructured":"Pelinovsky, E.N., Shurgalina, E.G.: Formation of freak waves in a soliton gas described by the modified Korteweg-de Vries equation. Doklady Phys. 61, 423\u2013426 (2016)","journal-title":"Doklady Phys."},{"key":"9559_CR34","doi-asserted-by":"publisher","first-page":"2049","DOI":"10.1016\/j.physleta.2016.04.023","volume":"380","author":"EG Shurgalina","year":"2016","unstructured":"Shurgalina, E.G., Pelinovsky, E.N.: Nonlinear dynamics of a soliton gas: modified Korteweg.de Vries equation framework. Phys. Lett. A 380, 2049\u20132053 (2016)","journal-title":"Phys. Lett. A"},{"key":"9559_CR35","doi-asserted-by":"publisher","first-page":"59","DOI":"10.1134\/S0015462818010123","volume":"53","author":"EG Shurgalina","year":"2018","unstructured":"Shurgalina, E.G.: Mechanism of the emergence of rogue waves as a result of the interaction between internal solitary waves in a stratified basin. Fluid Dyn. 53, 59\u201364 (2018)","journal-title":"Fluid Dyn."},{"key":"9559_CR36","doi-asserted-by":"publisher","first-page":"214501","DOI":"10.1103\/PhysRevLett.117.214501","volume":"117","author":"AV Slunyaev","year":"2016","unstructured":"Slunyaev, A.V., Pelinovsky, E.N.: Role of multiple soliton interactions in the generation of rogue waves: the modified Kortewegde Vries framework. Phys. Rev. Lett. 117, 214501 (2016)","journal-title":"Phys. Rev. Lett."},{"key":"9559_CR37","doi-asserted-by":"publisher","first-page":"330","DOI":"10.1063\/1.528449","volume":"30","author":"GZ Tu","year":"1989","unstructured":"Tu, G.Z.: The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J. Math. Phys. 30, 330\u2013338 (1989)","journal-title":"J. Math. Phys."},{"key":"9559_CR38","doi-asserted-by":"publisher","first-page":"904","DOI":"10.1137\/S0036139996303270","volume":"58","author":"E Varley","year":"1998","unstructured":"Varley, E., Seymour, B.R.: A simple derivation of the $$N$$-soliton solutions to the Korteweg-De Vries equation. SIAM J. Appl. Math. 58, 904\u2013911 (1998)","journal-title":"SIAM J. Appl. Math."},{"key":"9559_CR39","doi-asserted-by":"publisher","first-page":"435201","DOI":"10.1088\/1751-8113\/41\/43\/435201","volume":"41","author":"VM Vassilev","year":"2008","unstructured":"Vassilev, V.M., Djondjorov, P.A., Mladenov, I.M.: Cylindrical equilibrium shapes of fluid membranes. J. Phys. A: Math. Theor. 41, 435201 (2008)","journal-title":"J. Phys. A: Math. Theor."},{"key":"9559_CR40","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.physd.2016.03.002","volume":"321\u2013322","author":"OC Wright","year":"2016","unstructured":"Wright, O.C.: Effective integration of ultra-elliptic solutions of the focusing nonlinear Schr\u00f6odinger equation. Physica D 321\u2013322, 16\u201338 (2016)","journal-title":"Physica D"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-019-09559-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00332-019-09559-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-019-09559-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T00:05:06Z","timestamp":1694995506000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-019-09559-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,4]]},"references-count":40,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2019,12]]}},"alternative-id":["9559"],"URL":"https:\/\/doi.org\/10.1007\/s00332-019-09559-y","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,7,4]]},"assertion":[{"value":"9 August 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 June 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}