{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T18:44:11Z","timestamp":1790707451712,"version":"4.1.0"},"reference-count":62,"publisher":"The Royal Society","issue":"1867","license":[{"start":{"date-parts":[[2007,6,27]],"date-time":"2007-06-27T00:00:00Z","timestamp":1182902400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/royalsociety.org\/journals\/ethics-policies\/data-sharing-mining\/"}],"content-domain":{"domain":["royalsocietypublishing.org"],"crossmark-restriction":true},"short-container-title":["Phil. Trans. R. Soc. A."],"published-print":{"date-parts":[[2008,3,28]]},"abstract":"<jats:p>In the example of the Schr\u00f6dinger\/KdV equation, we treat the theory as equivalence of two concepts of Liouvillian integrability: quadrature integrability of linear differential equations with a parameter (spectral problem) and Liouville's integrability of finite-dimensional Hamiltonian systems (stationary KdV equations). Three key objects in this field\u2014new explicit<jats:italic>\u03a8<\/jats:italic>-function, trace formula and the Jacobi problem\u2014provide a complete solution. The<jats:italic>\u0398<\/jats:italic>-function language is derivable from these objects and used for ultimate representation of a solution to the inversion problem. Relations with non-integrable equations are also discussed.<\/jats:p>","DOI":"10.1098\/rsta.2007.2056","type":"journal-article","created":{"date-parts":[[2007,6,28]],"date-time":"2007-06-28T16:06:29Z","timestamp":1183046789000},"page":"923-945","update-policy":"https:\/\/doi.org\/10.1098\/crossmarkpolicy","source":"Crossref","is-referenced-by-count":10,"title":["What does integrability of finite-gap or soliton potentials mean?"],"prefix":"10.1098","volume":"366","author":[{"given":"Yurii V","family":"Brezhnev","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Boston UniversityBoston, MA 02215, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"175","published-online":{"date-parts":[[2007,6,27]]},"reference":[{"key":"e_1_3_3_2_1","doi-asserted-by":"crossref","unstructured":"Ablowitz M.J& Segur H Solitons and the inverse scattering transform. 1981 Philadelphia PA:SIAM.","DOI":"10.1137\/1.9781611970883"},{"key":"e_1_3_3_3_1","unstructured":"Acta Appl. Math . 1994 36 1\u2013308."},{"key":"e_1_3_3_4_1","first-page":"1409","article-title":"A continuous analogue to orthogonal polynomials on a system of intervals","volume":"2","author":"Akhiezer N.I","year":"1961","journal-title":"Soviet Math. Dokl"},{"key":"e_1_3_3_5_1","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-19.3.467"},{"key":"e_1_3_3_6_1","doi-asserted-by":"crossref","unstructured":"Baker H Abelian functions. Abel's theorem and the allied theory of theta functions. 1897 Cambridge UK:Cambridge University Press.","DOI":"10.3792\/chmm\/1428686976"},{"key":"e_1_3_3_7_1","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1928.0070"},{"key":"e_1_3_3_8_1","unstructured":"Belokolos E.D Bobenko A.I Enol'skii V.Z Its A.R& Matveev V.B Algebro-geometric approach to nonlinear integrable equations. 1994 Berlin Germany:Springer."},{"key":"e_1_3_3_9_1","doi-asserted-by":"publisher","DOI":"10.1070\/RM2002v057n02ABEH000503"},{"key":"e_1_3_3_10_1","unstructured":"Chudnovsky D. V. & Chudnovsky G. V. 1984 Appendix I: travaux de J. Drach (1919). In Classical and quantum models and ariphmetic problems (eds D. V. Chudnovsky & G. V. Chudnovsky). Lecture Notes in Pure and Applied Mathematics no. 92 pp. 445\u2013453. New York NY: Marcel Dekker."},{"key":"e_1_3_3_11_1","unstructured":"Clebsch A& Gordan P Theorie der Abelschen functionen. 1866 Leipzig Germany:Teubner."},{"key":"e_1_3_3_12_1","volume-title":"CRM series in mathematical physics","author":"Conte R","year":"1999"},{"key":"e_1_3_3_13_1","first-page":"1645","article-title":"Sur une \u00e9quation lin\u00e9are","author":"Darboux G","year":"1882","journal-title":"Compt. Rend. Acad. Sci"},{"key":"e_1_3_3_14_1","unstructured":"Darboux G. 1915 Lecons sur la Theorie Generale des Surfaces et les applications Geometriques du calcul infinitesimal . Deuxieme Parte 3rd edn. (Reprint.) New York NY: Chelsea Publications."},{"key":"e_1_3_3_15_1","first-page":"337","article-title":"Sur l'integration par quadratures de l'\u00e9quation diff\u00e9rentielle d2 y\/dx 2=[\u03c6(x)+h]y","volume":"168","author":"Drach J","year":"1919","journal-title":"Compt. Rend. Acad. Sci"},{"key":"e_1_3_3_16_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01078183"},{"key":"e_1_3_3_17_1","doi-asserted-by":"publisher","DOI":"10.1070\/RM1976v031n01ABEH001446"},{"key":"e_1_3_3_18_1","doi-asserted-by":"publisher","DOI":"10.1070\/RM1976v031n01ABEH001446"},{"key":"e_1_3_3_19_1","unstructured":"Eisenhart L.P Continuous groups of transformations. 1933 Princeton NJ:Princeton University Press."},{"key":"e_1_3_3_20_1","first-page":"1","article-title":"Differential equations of second order. Conditions of integrability in final form","volume":"9","author":"Ermakov V.P","year":"1880","journal-title":"Izvestiya Kievskogo Univ. III"},{"key":"e_1_3_3_21_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00250422"},{"key":"e_1_3_3_22_1","doi-asserted-by":"publisher","DOI":"10.1143\/PTP.55.438"},{"key":"e_1_3_3_23_1","doi-asserted-by":"publisher","DOI":"10.1063\/1.1665772"},{"key":"e_1_3_3_24_1","doi-asserted-by":"publisher","DOI":"10.1070\/RM1975v030n05ABEH001522"},{"key":"e_1_3_3_25_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01076434"},{"key":"e_1_3_3_26_1","doi-asserted-by":"crossref","unstructured":"Gesztesy F& Holden H Soliton equations and their algebro-geometric solutions. 1. 2003 Cambridge UK:Cambridge University Press.","DOI":"10.1017\/CBO9780511546723"},{"key":"e_1_3_3_27_1","doi-asserted-by":"crossref","unstructured":"Ibragimov N.H Transformation groups applied to mathematical physics. 1985 Dordrecht The Netherlands:D. Reidel Publishing.","DOI":"10.1007\/978-94-009-5243-0"},{"key":"e_1_3_3_28_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01078185"},{"key":"e_1_3_3_29_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01038218"},{"key":"e_1_3_3_30_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(86)80010-4"},{"key":"e_1_3_3_31_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01077946"},{"key":"e_1_3_3_32_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01135528"},{"key":"e_1_3_3_33_1","doi-asserted-by":"publisher","DOI":"10.1070\/RM1977v032n06ABEH003862"},{"key":"e_1_3_3_34_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01681429"},{"key":"e_1_3_3_35_1","doi-asserted-by":"crossref","unstructured":"Levitan B.M Inverse Sturm\u2013Liouville problems. 1987 Zeist The Netherlands:VSP.","DOI":"10.1515\/9783110941937"},{"key":"e_1_3_3_36_1","doi-asserted-by":"crossref","unstructured":"Levitan B. M. & Sargsjan I. S. 1975 Introduction to spectral theory: selfadjoint ordinary differential operators . Translations of mathematical monographs vol. 39. Providence RI: American Mathematical Society.","DOI":"10.1090\/mmono\/039"},{"key":"e_1_3_3_37_1","first-page":"423","article-title":"M\u00e9moire sur l'int\u00e9gration d'une classe d'\u00e9quations diff\u00e9rentielles du second ordre en quantit\u00e9s finies explicites","author":"Liouville J","year":"1839","journal-title":"Liouville's J"},{"key":"e_1_3_3_38_1","first-page":"1","article-title":"Remarques nouvelles sur l'\u00e9quation de Riccati","author":"Liouville J","year":"1841","journal-title":"Liouville's J"},{"key":"e_1_3_3_39_1","doi-asserted-by":"crossref","unstructured":"L\u00fctzen J Joseph Liouville 1809\u20131882. Master of pure and applied mathematics. 1990 New York NY:Springer.","DOI":"10.1007\/978-1-4612-0989-8"},{"key":"e_1_3_3_40_1","first-page":"1052","article-title":"The periodic Korteweg\u2013de Vries problem","volume":"15","author":"Marchenko V.A","year":"1974","journal-title":"Soviet Math. Dokl"},{"key":"e_1_3_3_41_1","first-page":"201","article-title":"New scheme of integration the KdV-equation","author":"Matveev V.B","year":"1975","journal-title":"Uspekhi Math. Nauk"},{"key":"e_1_3_3_42_1","unstructured":"Matveev V. B. 1976 Abelian functions and solitons. Preprint no. 373 pp. 1\u201398 Wroc\u0142aw University."},{"key":"e_1_3_3_43_1","doi-asserted-by":"crossref","unstructured":"Matveev V.B& Salle M.A Darboux transformations and solitons. 1991 Berlin Germany:Springer.","DOI":"10.1007\/978-3-662-00922-2"},{"key":"e_1_3_3_44_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01425567"},{"key":"e_1_3_3_45_1","volume-title":"Progress in mathematics","author":"Morales-Ruiz J.J","year":"1999"},{"key":"e_1_3_3_46_1","unstructured":"Mordukhai-Boltovskoi D. 1910 On the integration in finite terms of linear differential equations . Warsaw. [In Russian.]."},{"key":"e_1_3_3_47_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01075697"},{"key":"e_1_3_3_48_1","unstructured":"Novikov S.P. 1984 New York NY:Consultants Bureau."},{"key":"e_1_3_3_49_1","first-page":"34","article-title":"Note on a transformation of Riccati's equation","author":"Rawson R","year":"1883","journal-title":"Messenger Math"},{"key":"e_1_3_3_50_1","doi-asserted-by":"crossref","unstructured":"Ritt J.F Integration in finite terms. Liouville's theory of elementary methods. 1948 New York NY:Columbia University Press.","DOI":"10.7312\/ritt91596"},{"key":"e_1_3_3_51_1","doi-asserted-by":"publisher","DOI":"10.2307\/2374045"},{"key":"e_1_3_3_52_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02422445"},{"key":"e_1_3_3_53_1","first-page":"197","article-title":"Ueber das Umkehrproblem der elliptischen Integrale","author":"Tichomandritzky M","year":"1884","journal-title":"Math. Ann"},{"key":"e_1_3_3_54_1","unstructured":"Tikhomandritskii M. 1885 Inversion of hyperelliptic integrals. Doctoral thesis Khar'kov. [In Russian.]."},{"key":"e_1_3_3_55_1","unstructured":"Tikhomandritskii M. 1895 Foundations for the theory of Abelian integrals . Khar'kov. [In Russian.]Tikhomandritzky M. 1911 \u00c9l\u00e9ments de la th\u00e9orie des int\u00e9grales Ab\u00e9liennes Nouvelle \u00e9dition . St P\u00e9tersbourg Russia: Imprimerie de A. B\u00f6hnke. [In French.]."},{"key":"e_1_3_3_56_1","doi-asserted-by":"publisher","DOI":"10.1070\/RM2002v057n01ABEH000488"},{"key":"e_1_3_3_57_1","doi-asserted-by":"crossref","unstructured":"van der Put M& Singer M.F Galois theory of linear differential equations. 2003 Berlin Germany:Springer.","DOI":"10.1007\/978-3-642-55750-7"},{"key":"e_1_3_3_58_1","doi-asserted-by":"crossref","unstructured":"Vanhaecke P. 1996 Integrable systems in the realm of algebraic geometry vol. 1638. Lecture Notes in Mathematics. Berlin Germany: Springer.","DOI":"10.1007\/978-3-662-21535-7"},{"key":"e_1_3_3_59_1","doi-asserted-by":"crossref","unstructured":"Weierstrass K Theorie der Abel'schen Functionen. 1856 Berlin Germany:Druck und Verlag von Georg Reimer.","DOI":"10.1515\/crll.1856.52.285"},{"key":"e_1_3_3_60_1","unstructured":"Weierstrass K. 1902 Vorlesungen \u00fcber die theorie der Abeleschen transcendenten . Math. Werke. IV . Berlin Germany: Mayer & M\u00fcller."},{"key":"e_1_3_3_61_1","unstructured":"Whittaker E.T& Watson G.N A course of modern analysis. 4th edn. 1927 Cambridge UK:Cambridge University Press."},{"key":"e_1_3_3_62_1","unstructured":"Yakubovich V.A& Starzhinskii V.M Linear differential equations with periodic coefficients. 1 2. 1975 New York NY; Toronto Canada:Wiley."},{"key":"e_1_3_3_63_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01086739"}],"container-title":["Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/royalsocietypublishing.org\/doi\/pdf\/10.1098\/rsta.2007.2056","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/royalsocietypublishing.org\/doi\/full-xml\/10.1098\/rsta.2007.2056","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/royalsocietypublishing.org\/doi\/pdf\/10.1098\/rsta.2007.2056","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,12]],"date-time":"2023-05-12T18:44:01Z","timestamp":1683917041000},"score":1,"resource":{"primary":{"URL":"https:\/\/royalsocietypublishing.org\/doi\/10.1098\/rsta.2007.2056"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,6,27]]},"references-count":62,"journal-issue":{"issue":"1867","published-print":{"date-parts":[[2008,3,28]]}},"alternative-id":["10.1098\/rsta.2007.2056"],"URL":"https:\/\/doi.org\/10.1098\/rsta.2007.2056","relation":{},"ISSN":["1364-503X","1471-2962"],"issn-type":[{"value":"1364-503X","type":"print"},{"value":"1471-2962","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,6,27]]}}}