{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T02:54:46Z","timestamp":1772765686035,"version":"3.50.1"},"reference-count":73,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T00:00:00Z","timestamp":1724889600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T00:00:00Z","timestamp":1724889600000},"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":"publisher","id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004739","name":"Youth Innovation Promotion Association of the Chinese Academy of Sciences","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004739","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004608","name":"Natural Science Foundation of Jiangsu Province","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004608","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2024,10]]},"DOI":"10.1007\/s00332-024-10072-0","type":"journal-article","created":{"date-parts":[[2024,8,29]],"date-time":"2024-08-29T19:02:16Z","timestamp":1724958136000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Integrable Variants of the Toda Lattice"],"prefix":"10.1007","volume":"34","author":[{"given":"Ya-Jie","family":"Liu","sequence":"first","affiliation":[]},{"given":"Hui Alan","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Xiang-Ke","family":"Chang","sequence":"additional","affiliation":[]},{"given":"Xing-Biao","family":"Hu","sequence":"additional","affiliation":[]},{"given":"Ying-Nan","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,29]]},"reference":[{"key":"10072_CR1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511998324","volume-title":"Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons","author":"MJ Ablowitz","year":"2011","unstructured":"Ablowitz, M.J.: Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons. Cambridge University Press, Cambridge (2011)"},{"key":"10072_CR2","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511623998","volume-title":"Nonlinear Evolution Equation and Inverse Scattering Solitons","author":"MJ Ablowitz","year":"1991","unstructured":"Ablowitz, M.J., Clarkson, P.: Nonlinear Evolution Equation and Inverse Scattering Solitons. Cambridge University Press, Cambridge (1991)"},{"key":"10072_CR3","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.110.064105","volume":"110","author":"MJ Ablowitz","year":"2013","unstructured":"Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schr\u00f6dinger equation. Phys. Rev. Lett. 110, 064105 (2013)","journal-title":"Phys. Rev. Lett."},{"issue":"10","key":"10072_CR4","doi-asserted-by":"crossref","first-page":"2180","DOI":"10.1063\/1.523550","volume":"19","author":"MJ Ablowitz","year":"1978","unstructured":"Ablowitz, M.J., Satsuma, J.: Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys. 19(10), 2180\u20132186 (1978)","journal-title":"J. Math. Phys."},{"key":"10072_CR5","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970883","volume-title":"Solitons and The Inverse Scattering Transform","author":"MJ Ablowitz","year":"1981","unstructured":"Ablowitz, M.J., Segur, H.: Solitons and The Inverse Scattering Transform. SIAM, London (1981)"},{"key":"10072_CR6","volume-title":"Discrete and Continuous Nonlinear Schr\u00f6dinger Systems","author":"MJ Ablowitz","year":"2004","unstructured":"Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and Continuous Nonlinear Schr\u00f6dinger Systems. Cambridge University Press, Cambridge (2004)"},{"key":"10072_CR7","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1215\/S0012-7094-95-08029-6","volume":"80","author":"M Adler","year":"1995","unstructured":"Adler, M., Van Moerbeke, P.: Matrix integrals, Toda symmetries, Virasoro constraints and orthogonal polynomials. Duke Math. J. 80, 863\u2013911 (1995)","journal-title":"Duke Math. J."},{"key":"10072_CR8","volume-title":"Solitons: Nonlinear Pulses and Beams","author":"NN Akhmediev","year":"1997","unstructured":"Akhmediev, N.N., Ankiewicz, A.: Solitons: Nonlinear Pulses and Beams. Chapman & Hall, London (1997)"},{"issue":"6","key":"10072_CR9","doi-asserted-by":"crossref","first-page":"675","DOI":"10.1016\/j.physleta.2008.12.036","volume":"373","author":"N Akhmediev","year":"2009","unstructured":"Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373(6), 675\u2013678 (2009)","journal-title":"Phys. Lett. A"},{"issue":"17","key":"10072_CR10","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.116.173901","volume":"116","author":"F Baronio","year":"2016","unstructured":"Baronio, F., Wabnitz, S., Kodama, Y.: Optical Kerr spatiotemporal dark-lump dynamics of hydrodynamic origin. Phys. Rev. Lett. 116(17), 173901 (2016)","journal-title":"Phys. Rev. Lett."},{"issue":"1","key":"10072_CR11","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1002\/1097-0312(200101)54:1<91::AID-CPA4>3.0.CO;2-O","volume":"54","author":"R Beals","year":"2001","unstructured":"Beals, R., Sattinger, D.H., Szmigielski, J.: Peakons, strings, and the finite Toda lattice. Commun. Pure Appl. Math. 54(1), 91\u2013106 (2001)","journal-title":"Commun. Pure Appl. Math."},{"key":"10072_CR12","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1103\/PhysRevLett.84.745","volume":"84","author":"P Binder","year":"2000","unstructured":"Binder, P., Abraimov, D., Ustinov, A.V., Flach, S., Zolotaryuk, Y.: Observation of breathers in Josephson ladders. Phys. Rev. Lett. 84, 745\u2013748 (2000)","journal-title":"Phys. Rev. Lett."},{"issue":"6","key":"10072_CR13","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.99.064103","volume":"99","author":"G Biondini","year":"2007","unstructured":"Biondini, G.: Line soliton interactions of the Kadomtsev-Petviashvili equation. Phys. Rev. Lett. 99(6), 064103 (2007)","journal-title":"Phys. Rev. Lett."},{"issue":"43","key":"10072_CR14","doi-asserted-by":"crossref","DOI":"10.1088\/1751-8113\/43\/43\/434007","volume":"43","author":"G Biondini","year":"2010","unstructured":"Biondini, G., Wang, D.: On the soliton solutions of the two-dimensional Toda lattice. J. Phys. A Math. Theor. 43(43), 434007 (2010)","journal-title":"J. Phys. A Math. Theor."},{"key":"10072_CR15","doi-asserted-by":"crossref","first-page":"1661","DOI":"10.1103\/PhysRevLett.71.1661","volume":"71","author":"R Camassa","year":"1993","unstructured":"Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661\u20131664 (1993)","journal-title":"Phys. Rev. Lett."},{"issue":"8","key":"10072_CR16","doi-asserted-by":"crossref","first-page":"3562","DOI":"10.1073\/pnas.75.8.3562","volume":"75","author":"KM Case","year":"1978","unstructured":"Case, K.M.: The $$N$$-soliton solution of the Benjamin-Ono equation. Proc. Nat. Acad. Sci. U.S.A. 75(8), 3562\u20133563 (1978)","journal-title":"Proc. Nat. Acad. Sci. U.S.A."},{"issue":"6","key":"10072_CR17","doi-asserted-by":"crossref","first-page":"2797","DOI":"10.1007\/s00332-019-09559-y","volume":"29","author":"J Chen","year":"2019","unstructured":"Chen, J., Pelinovsky, D.E.: Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background. J. Nonlinear Sci. 29(6), 2797\u20132843 (2019)","journal-title":"J. Nonlinear Sci."},{"key":"10072_CR18","doi-asserted-by":"crossref","unstructured":"Faddeev, L., Van Moerbeke, P., Lambert, F. (eds.): Bilinear Integrable Systems: From Classical to Quantum, Continuous to Discrete. NATO Science Series II: Mathematics, vol. 201. Physics and Chemistry. Springer, Dordrecht (2006)","DOI":"10.1007\/978-1-4020-3503-6"},{"issue":"1","key":"10072_CR19","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1111\/sapm.12287","volume":"144","author":"B-F Feng","year":"2020","unstructured":"Feng, B.-F., Ling, L., Takahashi, D.A.: Multi-breather and high-order rogue waves for the nonlinear Schr\u00f6dinger equation on the elliptic function background. Stud. Appl. Math. 144(1), 46\u2013101 (2020)","journal-title":"Stud. Appl. Math."},{"issue":"1\u20133","key":"10072_CR20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2008.05.002","volume":"467","author":"S Flach","year":"2008","unstructured":"Flach, S., Gorbach, A.V.: Discrete breathers-advances in theory and applications. Phys. Rep. 467(1\u20133), 1\u2013116 (2008)","journal-title":"Phys. Rep."},{"issue":"5","key":"10072_CR21","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/S0370-1573(97)00068-9","volume":"295","author":"S Flach","year":"1998","unstructured":"Flach, S., Willis, C.R.: Discrete breathers. Phys. Rep. 295(5), 181\u2013264 (1998)","journal-title":"Phys. Rep."},{"key":"10072_CR22","doi-asserted-by":"crossref","unstructured":"Gegenhasi, X.\u00a0B. Hu, S.\u00a0H. Li, and B.\u00a0Wang. Nonsingular rational solutions to integrable models. In Asymptotic, Algebraic and Geometric Aspects of Integrable Systems, pages 79\u201399. Springer, 2020","DOI":"10.1007\/978-3-030-57000-2_6"},{"issue":"42","key":"10072_CR23","doi-asserted-by":"crossref","first-page":"12619","DOI":"10.1088\/1751-8113\/40\/42\/S08","volume":"40","author":"B Grammaticos","year":"2007","unstructured":"Grammaticos, B., Ramani, A., Papageorgiou, V., Satsuma, J., Willox, R.: Constructing lump-like solutions of the Hirota-Miwa equation. J. Phys. A Math. Theor. 40(42), 12619 (2007)","journal-title":"J. Phys. A Math. Theor."},{"key":"10072_CR24","doi-asserted-by":"crossref","unstructured":"Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, 2004. Translated by Nagai, A., Nimmo, J. and Gilson, C","DOI":"10.1017\/CBO9780511543043"},{"key":"10072_CR25","doi-asserted-by":"crossref","first-page":"1192","DOI":"10.1103\/PhysRevLett.27.1192","volume":"27","author":"R Hirota","year":"1971","unstructured":"Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192\u20131194 (1971)","journal-title":"Phys. Rev. Lett."},{"issue":"1","key":"10072_CR26","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1143\/JPSJ.35.286","volume":"35","author":"R Hirota","year":"1973","unstructured":"Hirota, R.: Exact $$N$$-soliton solution of a nonlinear lumped network equation. J. Phys. Soc. Japan 35(1), 286\u2013288 (1973)","journal-title":"J. Phys. Soc. Japan"},{"issue":"1","key":"10072_CR27","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1143\/JPSJ.50.338","volume":"50","author":"R Hirota","year":"1981","unstructured":"Hirota, R., Ito, M.: A direct approach to multi-periodic wave solutions to nonlinear evolution equations. J. Phys. Soc. Japan 50(1), 338\u2013342 (1981)","journal-title":"J. Phys. Soc. Japan"},{"key":"10072_CR28","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1143\/PTPS.59.64","volume":"59","author":"R Hirota","year":"1976","unstructured":"Hirota, R., Satsuma, J.: A variety of nonlinear network equations generated from the B\u00e4cklund transformation for the Toda lattice. Prog. Theor. Phys. Supp. 59, 64\u2013100 (1976)","journal-title":"Prog. Theor. Phys. Supp."},{"issue":"24","key":"10072_CR29","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevLett.118.243903","volume":"118","author":"TP Horikis","year":"2017","unstructured":"Horikis, T.P., Frantzeskakis, D.J.: Light meets water in nonlocal media: surface tension analogue in optics. Phys. Rev. Lett. 118(24), 243903 (2017)","journal-title":"Phys. Rev. Lett."},{"issue":"5","key":"10072_CR30","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","volume":"2","author":"K Hornik","year":"1989","unstructured":"Hornik, K., Stinchcombe, M., White, H.: Multilayer feedforward networks are universal approximators. Neural Netw. 2(5), 359\u2013366 (1989)","journal-title":"Neural Netw."},{"issue":"1","key":"10072_CR31","first-page":"201","volume":"27","author":"XB Hu","year":"1994","unstructured":"Hu, X.B.: Nonlinear superposition formulae for the differential-difference analogue of the KdV equation and two-dimensional Toda equation. J. Phys. A Math. Theor. 27(1), 201\u2013214 (1994)","journal-title":"J. Phys. A Math. Theor."},{"key":"10072_CR32","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139171281","volume-title":"Nonlinear Waves, Solitons and Chaos","author":"E Infeld","year":"2000","unstructured":"Infeld, E., Rowlands, G.: Nonlinear Waves, Solitons and Chaos, 2nd edn. Cambridge University Press, Cambridge (2000)","edition":"2"},{"key":"10072_CR33","first-page":"539","volume":"15","author":"BB Kadomtsev","year":"1970","unstructured":"Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539\u2013541 (1970)","journal-title":"Sov. Phys. Dokl."},{"issue":"4","key":"10072_CR34","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1103\/RevModPhys.61.763","volume":"61","author":"YS Kivshar","year":"1989","unstructured":"Kivshar, Y.S., Malomed, B.A.: Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61(4), 763 (1989)","journal-title":"Rev. Mod. Phys."},{"key":"10072_CR35","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611975529","volume-title":"Solitons in Two-Dimensional Shallow Water","author":"Y Kodama","year":"2018","unstructured":"Kodama, Y.: Solitons in Two-Dimensional Shallow Water. SIAM, London (2018)"},{"issue":"3","key":"10072_CR36","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/0001-8708(79)90057-4","volume":"34","author":"B Kostant","year":"1979","unstructured":"Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34(3), 195\u2013338 (1979)","journal-title":"Adv. Math."},{"key":"10072_CR37","volume-title":"Elements of Soliton Theory","author":"GL Lamb Jr","year":"1980","unstructured":"Lamb, G.L., Jr.: Elements of Soliton Theory. Wiley, London (1980)"},{"issue":"7","key":"10072_CR38","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1023\/A:1006910004292","volume":"30","author":"KE Lonngren","year":"1998","unstructured":"Lonngren, K.E.: Ion acoustic soliton experiments in a plasma. Opt. Quant. Electron. 30(7), 615\u2013630 (1998)","journal-title":"Opt. Quant. Electron."},{"issue":"3","key":"10072_CR39","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/0375-9601(77)90875-1","volume":"63","author":"SV Manakov","year":"1977","unstructured":"Manakov, S.V., Zakharov, V.E., Bordag, L.A., Its, A.R., Matveev, V.B.: Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction. Phys. Lett. A 63(3), 205\u2013206 (1977)","journal-title":"Phys. Lett. A"},{"issue":"3","key":"10072_CR40","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1007\/BF02102036","volume":"138","author":"EJ Martinec","year":"1991","unstructured":"Martinec, E.J.: On the origin of integrability in matrix models. Commun. Math. Phys. 138(3), 437\u2013449 (1991)","journal-title":"Commun. Math. Phys."},{"issue":"5","key":"10072_CR41","doi-asserted-by":"crossref","first-page":"1701","DOI":"10.1143\/JPSJ.47.1701","volume":"47","author":"A Nakamura","year":"1979","unstructured":"Nakamura, A.: A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. exact two-periodic wave solution. J. Phys. Soc. Japan 47(5), 1701\u20131705 (1979)","journal-title":"J. Phys. Soc. Japan"},{"issue":"2","key":"10072_CR42","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1143\/JPSJ.52.380","volume":"52","author":"A Nakamura","year":"1983","unstructured":"Nakamura, A.: Exact Bessel type solution of the two-dimensional Toda lattice equation. J. Phys. Soc. Japan 52(2), 380\u2013387 (1983)","journal-title":"J. Phys. Soc. Japan"},{"issue":"10","key":"10072_CR43","doi-asserted-by":"crossref","first-page":"3309","DOI":"10.1143\/JPSJ.57.3309","volume":"57","author":"A Nakamura","year":"1988","unstructured":"Nakamura, A.: Exact cylindrical soliton solutions of the sine-Gordon equation, the sinh-Gordon equation and the periodic Toda equation. J. Phys. Soc. Japan 57(10), 3309\u20133322 (1988)","journal-title":"J. Phys. Soc. Japan"},{"issue":"2","key":"10072_CR44","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1016\/S0022-247X(03)00124-0","volume":"281","author":"K Narita","year":"2003","unstructured":"Narita, K.: Rational and $$N$$-breather solutions for the 2D Toda lattice equation. J. Math. Anal. Appl. 281(2), 757\u2013760 (2003)","journal-title":"J. Math. Anal. Appl."},{"key":"10072_CR45","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970227","volume-title":"Solitons in Mathematics and Physics","author":"AC Newell","year":"1985","unstructured":"Newell, A.C.: Solitons in Mathematics and Physics. SIAM, London (1985)"},{"issue":"3","key":"10072_CR46","doi-asserted-by":"crossref","first-page":"749","DOI":"10.1143\/JPSJ.52.749","volume":"52","author":"K Ohkuma","year":"1983","unstructured":"Ohkuma, K., Wadati, M.: The Kadomtsev-Petviashvili equation: the trace method and the soliton resonances. J. Phys. Soc. Japan 52(3), 749\u2013760 (1983)","journal-title":"J. Phys. Soc. Japan"},{"issue":"5","key":"10072_CR47","doi-asserted-by":"crossref","first-page":"5016","DOI":"10.1103\/PhysRevE.51.5016","volume":"51","author":"DE Pelinovsky","year":"1995","unstructured":"Pelinovsky, D.E., Stepanyants, Y.A., Kivshar, Y.S.: Self-focusing of plane dark solitons in nonlinear defocusing media. Phys. Rev. E 51(5), 5016 (1995)","journal-title":"Phys. Rev. E"},{"key":"10072_CR48","first-page":"469","volume":"2","author":"VI Petviashvili","year":"1976","unstructured":"Petviashvili, V.I.: Equation of an extraordinary soliton. Plasma Phys. 2, 469\u2013472 (1976)","journal-title":"Plasma Phys."},{"issue":"4","key":"10072_CR49","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s00332-021-09720-6","volume":"31","author":"J Rao","year":"2021","unstructured":"Rao, J., Fokas, A.S., He, J.: Doubly localized two-dimensional rogue waves in the Davey-Stewartson I equation. J. Nonlinear Sci. 31(4), 67 (2021)","journal-title":"J. Nonlinear Sci."},{"issue":"2","key":"10072_CR50","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1007\/BF02097366","volume":"133","author":"SNM Ruijsenaars","year":"1990","unstructured":"Ruijsenaars, S.N.M.: Relativistic Toda systems. Commun. Math. Phys. 133(2), 217\u2013247 (1990)","journal-title":"Commun. Math. Phys."},{"issue":"16","key":"10072_CR51","doi-asserted-by":"crossref","first-page":"1798","DOI":"10.1103\/PhysRevLett.59.1798","volume":"59","author":"S Saito","year":"1987","unstructured":"Saito, S.: String theories and Hirota\u2019s bilinear difference equation. Phys. Rev. Lett. 59(16), 1798 (1987)","journal-title":"Phys. Rev. Lett."},{"issue":"1","key":"10072_CR52","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1143\/JPSJ.49.409","volume":"49","author":"N Saitoh","year":"1980","unstructured":"Saitoh, N.: A transformation connecting the Toda lattice and the K-dV equation. J. Phys. Soc. Japan 49(1), 409\u2013416 (1980)","journal-title":"J. Phys. Soc. Japan"},{"issue":"7","key":"10072_CR53","doi-asserted-by":"crossref","first-page":"1496","DOI":"10.1063\/1.524208","volume":"20","author":"J Satsuma","year":"1979","unstructured":"Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20(7), 1496\u20131503 (1979)","journal-title":"J. Math. Phys."},{"key":"10072_CR54","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1103\/PhysRevLett.83.223","volume":"83","author":"UT Schwarz","year":"1999","unstructured":"Schwarz, U.T., English, L.Q., Sievers, A.J.: Experimental generation and observation of intrinsic localized spin wave modes in an antiferromagnet. Phys. Rev. Lett. 83, 223\u2013226 (1999)","journal-title":"Phys. Rev. Lett."},{"key":"10072_CR55","doi-asserted-by":"crossref","first-page":"970","DOI":"10.1103\/PhysRevLett.61.970","volume":"61","author":"AJ Sievers","year":"1988","unstructured":"Sievers, A.J., Takeno, S.: Intrinsic localized modes in anharmonic crystals. Phys. Rev. Lett. 61, 970\u2013973 (1988)","journal-title":"Phys. Rev. Lett."},{"issue":"7172","key":"10072_CR56","doi-asserted-by":"crossref","first-page":"1054","DOI":"10.1038\/nature06402","volume":"450","author":"DR Solli","year":"2007","unstructured":"Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450(7172), 1054\u20131057 (2007)","journal-title":"Nature"},{"issue":"1","key":"10072_CR57","doi-asserted-by":"crossref","first-page":"110","DOI":"10.1137\/S003614459631241X","volume":"40","author":"W Squire","year":"1998","unstructured":"Squire, W., Trapp, G.: Using complex variables to estimate derivatives of real functions. SIAM Rev. 40(1), 110\u2013112 (1998)","journal-title":"SIAM Rev."},{"issue":"9","key":"10072_CR58","doi-asserted-by":"crossref","first-page":"L499","DOI":"10.1088\/0305-4470\/21\/9\/004","volume":"21","author":"L Stenflo","year":"1988","unstructured":"Stenflo, L.: A solution of the generalised non-linear Schr\u00f6dinger equation. J. Phys. A Math. Gen. 21(9), L499 (1988)","journal-title":"J. Phys. A Math. Gen."},{"issue":"1","key":"10072_CR59","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1109\/JQE.2002.806184","volume":"39","author":"AA Sukhorukov","year":"2003","unstructured":"Sukhorukov, A.A., Kivshar, Y.S., Eisenberg, H.S., Silberberg, Y.: Spatial optical solitons in waveguide arrays. IEEE J. Quantum Electron. 39(1), 31\u201350 (2003)","journal-title":"IEEE J. Quantum Electron."},{"issue":"1","key":"10072_CR60","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1007\/BF02101675","volume":"181","author":"K Takasaki","year":"1996","unstructured":"Takasaki, K.: Toda lattice hierarchy and generalized string equations. Commun. Math. Phys. 181(1), 131\u2013156 (1996)","journal-title":"Commun. Math. Phys."},{"key":"10072_CR61","unstructured":"M.\u00a0Toda. Selected Papers of Morikazu Toda. World Scientific, 1993. ed. by M. Wadati"},{"issue":"2","key":"10072_CR62","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1143\/JPSJ.22.431","volume":"22","author":"M Toda","year":"1967","unstructured":"Toda, M.: Vibration of a chain with nonlinear interaction. J. Phys. Soc. Japan 22(2), 431\u2013436 (1967)","journal-title":"J. Phys. Soc. Japan"},{"key":"10072_CR63","volume-title":"Nonlinear Waves and Solitons","author":"M Toda","year":"1989","unstructured":"Toda, M.: Nonlinear Waves and Solitons. Springer, Berlin (1989)"},{"key":"10072_CR64","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-83219-2","volume-title":"Theory of Nonlinear Lattices","author":"M Toda","year":"1989","unstructured":"Toda, M.: Theory of Nonlinear Lattices, 2nd edn. Springer, Berlin (1989)","edition":"2"},{"key":"10072_CR65","doi-asserted-by":"crossref","first-page":"741","DOI":"10.1103\/PhysRevLett.84.741","volume":"84","author":"E Tr\u00edas","year":"2000","unstructured":"Tr\u00edas, E., Mazo, J.J., Orlando, T.P.: Discrete breathers in nonlinear lattices: Experimental detection in a Josephson array. Phys. Rev. Lett. 84, 741\u2013744 (2000)","journal-title":"Phys. Rev. Lett."},{"key":"10072_CR66","unstructured":"K.\u00a0Ueno and K.\u00a0Takasaki. Toda lattice hierarchy. In Advanced Studies in Pure Mathematics, vol. 9, pp. 1\u201395. North-Holland, Amsterdam, 1984"},{"issue":"1","key":"10072_CR67","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1137\/S0036139996300903","volume":"59","author":"J Villarroel","year":"1998","unstructured":"Villarroel, J.: On the solution to the inverse problem for the Toda chain. SIAM J. Appl. Math. 59(1), 261\u2013285 (1998)","journal-title":"SIAM J. Appl. Math."},{"issue":"1","key":"10072_CR68","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s002200050716","volume":"207","author":"J Villarroel","year":"1999","unstructured":"Villarroel, J., Ablowitz, M.J.: On the discrete spectrum of the nonstationary Schr\u00f6dinger equation and multipole lumps of the Kadomtsev-Petviashvili I equation. Commun. Math. Phys. 207(1), 1\u201342 (1999)","journal-title":"Commun. Math. Phys."},{"key":"10072_CR69","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719680","volume-title":"Nonlinear Waves in Integrable and Nonintegrable Systems","author":"J Yang","year":"2010","unstructured":"Yang, J.: Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM, London (2010)"},{"issue":"4","key":"10072_CR70","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1007\/s00332-022-09807-8","volume":"32","author":"B Yang","year":"2022","unstructured":"Yang, B., Yang, J.: Pattern transformation in higher-order lumps of the Kadomtsev-Petviashvili I equation. J. Nonlinear Sci. 32(4), 52 (2022)","journal-title":"J. Nonlinear Sci."},{"issue":"6","key":"10072_CR71","doi-asserted-by":"crossref","first-page":"3027","DOI":"10.1007\/s00332-020-09643-8","volume":"30","author":"B Yang","year":"2020","unstructured":"Yang, B., Chen, J., Yang, J.: Rogue waves in the generalized derivative nonlinear Schr\u00f6dinger equations. J. Nonlinear Sci. 30(6), 3027\u20133056 (2020)","journal-title":"J. Nonlinear Sci."},{"key":"10072_CR72","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1016\/j.jcp.2017.11.027","volume":"355","author":"YN Zhang","year":"2018","unstructured":"Zhang, Y.N., Hu, X.B., Sun, J.Q.: A numerical study of the 3-periodic wave solutions to KdV-type equations. J. Comput. Phys. 355, 566\u2013581 (2018)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"10072_CR73","doi-asserted-by":"crossref","first-page":"579","DOI":"10.4208\/cicp.OA-2018-0157","volume":"26","author":"YN Zhang","year":"2019","unstructured":"Zhang, Y.N., Hu, X.B., He, Y., Sun, J.Q.: A numerical study of the 3-periodic wave solutions to Toda-type equations. Commun. Comput. Phys. 26(2), 579\u2013598 (2019)","journal-title":"Commun. Comput. Phys."}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-024-10072-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-024-10072-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-024-10072-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,5]],"date-time":"2024-09-05T04:06:42Z","timestamp":1725509202000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-024-10072-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,29]]},"references-count":73,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["10072"],"URL":"https:\/\/doi.org\/10.1007\/s00332-024-10072-0","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,29]]},"assertion":[{"value":"2 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 July 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 August 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"98"}}