{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T11:54:52Z","timestamp":1763034892835,"version":"3.41.2"},"reference-count":33,"publisher":"AIP Publishing","issue":"9","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2008,9,1]]},"abstract":"<jats:p>We study the Harry Dym hierarchy of nonlinear evolution equations from the bi-Hamiltonian view point. This is done by using the concept of an S-hierarchy. It allows us to define a matrix Harry Dym hierarchy of commuting Hamiltonian flows in two fields that projects onto the scalar Harry Dym hierarchy. We also show that the conserved densities of the matrix Harry Dym equation can be found by means of a Riccati-type equation.<\/jats:p>","DOI":"10.1063\/1.2982232","type":"journal-article","created":{"date-parts":[[2008,9,26]],"date-time":"2008-09-26T12:37:39Z","timestamp":1222432659000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Bi-Hamiltonian aspects of a matrix Harry Dym hierarchy"],"prefix":"10.1063","volume":"49","author":[{"given":"Laura","family":"Fontanelli","sequence":"first","affiliation":[{"name":"Universit\u00e0 di Milano-Bicocca 1 Dipartimento di Matematica e Applicazioni, , Via Roberto Cozzi 53, I-20125 Milano, Italy"}]},{"given":"Paolo","family":"Lorenzoni","sequence":"additional","affiliation":[{"name":"Universit\u00e0 di Milano-Bicocca 1 Dipartimento di Matematica e Applicazioni, , Via Roberto Cozzi 53, I-20125 Milano, Italy"}]},{"given":"Marco","family":"Pedroni","sequence":"additional","affiliation":[{"name":"Universit\u00e0 di Bergamo 2 Dipartimento di Ingegneria dell\u2019Informazione e Metodi Matematici, , Viale Marconi 5, I-24044 Dalmine (BG), Italy"}]},{"given":"Jorge P.","family":"Zubelli","sequence":"additional","affiliation":[{"name":"IMPA 3 , Est. D. Castorina 110, Rio de Janeiro, Rio de Janeiro 22460-320, Brazil"}]}],"member":"317","published-online":{"date-parts":[[2008,9,25]]},"reference":[{"key":"2023080108475977000_c1","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1016\/0167-2789(87)90023-6","article-title":"Coupled KdV equations with multi-Hamiltonian structures","volume":"28","year":"1987","journal-title":"Physica D"},{"key":"2023080108475977000_c2","doi-asserted-by":"publisher","first-page":"L269","DOI":"10.1088\/0305-4470\/21\/5\/001","article-title":"Coupled Harry Dym equations with multi-Hamiltonian structures","volume":"21","year":"1988","journal-title":"J. Phys. A"},{"key":"2023080108475977000_c3","doi-asserted-by":"publisher","first-page":"1099","DOI":"10.1088\/0305-4470\/39\/5\/006","article-title":"On negative flows of the AKNS hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type","volume":"39","year":"2006","journal-title":"J. Phys. A"},{"volume-title":"Spectral Transform and Solitons","year":"1982","key":"2023080108475977000_c4"},{"key":"2023080108475977000_c5","first-page":"45","article-title":"Darboux transformations and Darboux coverings: Some applications to the KP hierarchy","volume":"15","year":"1998","journal-title":"Contemp. Math."},{"key":"2023080108475977000_c6","first-page":"251","volume-title":"Integrable Systems\u2013Luminy 1991","author":"Babelon","year":"1993"},{"key":"2023080108475977000_c7","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/BF00398305","article-title":"Drinfeld\u2013Sokolov reduction on a simple lie algebra from the bihamiltonian point of view","volume":"25","year":"1992","journal-title":"Lett. Math. Phys."},{"key":"2023080108475977000_c8","doi-asserted-by":"publisher","first-page":"1975","DOI":"10.1007\/BF02105860","article-title":"Lie algebras and equations of Korteweg\u2013de Vries type","volume":"30","year":"1985","journal-title":"J. Sov. Math."},{"key":"2023080108475977000_c9","doi-asserted-by":"publisher","first-page":"4619","DOI":"10.1088\/0305-4470\/27\/13\/035","article-title":"\u2202\u00af-dressing and exact solutions for the (2+1)-dimensional Harry Dym equation","volume":"27","year":"1994","journal-title":"J. Phys. A"},{"key":"2023080108475977000_c10","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s002200050452","article-title":"Bihamiltonian geometry, Darboux coverings, and linearization of the KP hierarchy","volume":"197","year":"1998","journal-title":"Commun. Math. Phys."},{"key":"2023080108475977000_c11","doi-asserted-by":"publisher","first-page":"1150","DOI":"10.1063\/1.1542921","article-title":"Reciprocal transformations of Hamiltonian operators of hydrodynamic type: Nonlocal Hamiltonian formalism for linearly degenerate systems","volume":"44","year":"2003","journal-title":"J. Math. Phys."},{"key":"2023080108475977000_c12","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s11005-006-0109-z","article-title":"A three-component extension of the Camassa-Holm hierarchy","volume":"78","year":"2006","journal-title":"Lett. Math. Phys."},{"key":"2023080108475977000_c13","first-page":"283","article-title":"On Harry-Dym equation and its solution","volume":"32","year":"1989","journal-title":"Sci. China, Ser. A: Math., Phys., Astron. Technol. Sci."},{"key":"2023080108475977000_c14","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1007\/BF01016150","article-title":"The Harry Dym equation on the complex plane: Inverse spectral transform and exact solutions","volume":"99","year":"1994","journal-title":"Theor. Math. Phys."},{"key":"2023080108475977000_c15","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1016\/0167-2789(94)00153-H","article-title":"Inverse spectral transform for the Harry Dym equation on the complex plane","volume":"81","year":"1995","journal-title":"Physica D"},{"first-page":"310","volume-title":"Dynamical Systems, Theory and Applications","year":"1975","key":"2023080108475977000_c16"},{"volume-title":"Symplectic Geometry and Analytical Mechanics","year":"1987","key":"2023080108475977000_c17"},{"key":"2023080108475977000_c18","doi-asserted-by":"crossref","first-page":"4019","DOI":"10.1155\/S1073792804142554","article-title":"On the bihamiltonian structures of the Camassa-Holm and Harry Dym equations","volume":"2004","year":"2004","journal-title":"Int. Math. Res. Notices"},{"key":"2023080108475977000_c19","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/b94605","volume-title":"Integrability of Nonlinear Systems","author":"Kosmann-Schwarzbach","year":"2004","edition":"2nd ed."},{"key":"2023080108475977000_c20","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/s002200050166","article-title":"On the geometry of Darboux transformations for the KP hierarchy and its connection with the discrete KP hierarchy","volume":"188","year":"1997","journal-title":"Commun. Math. Phys."},{"key":"2023080108475977000_c21","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1006\/jmaa.1997.5707","article-title":"Squared eigenfunction symmetries for soliton equations: I, II","volume":"217","year":"1998","journal-title":"J. Math. Anal. Appl."},{"key":"2023080108475977000_c22","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1142\/S0129055X93000073","article-title":"Gauge transformations and reciprocal links in 2+1 dimensions","volume":"5","year":"1993","journal-title":"Rev. Math. Phys."},{"key":"2023080108475977000_c23","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1007\/BF00750836","article-title":"Equivalence of the Drinfeld-Sokolov reduction to a bihamiltonian reduction","volume":"35","year":"1995","journal-title":"Lett. Math. Phys."},{"key":"2023080108475977000_c24","unstructured":"Pedroni, M., \u201cA bihamiltonian view on the Boussinesq hierarchy,\u201d Universit\u00e0 di Genova Preprint No. 280, 1995."},{"key":"2023080108475977000_c25","first-page":"1583","article-title":"On the bihamiltonian theory for the Harry Dym equation","volume":"133","year":"2002","journal-title":"Theor. Math. Phys."},{"key":"2023080108475977000_c26","doi-asserted-by":"publisher","first-page":"260","DOI":"10.1016\/j.physleta.2003.08.037","article-title":"The generalized Harry Dym equation","volume":"317","year":"2003","journal-title":"Phys. Lett. A"},{"key":"2023080108475977000_c27","doi-asserted-by":"publisher","first-page":"834","DOI":"10.1063\/1.530616","article-title":"Integrable mechanical systems related to the Harry-Dym hierarchy","volume":"35","year":"1994","journal-title":"J. Math. Phys."},{"key":"2023080108475977000_c28","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/0375-9601(87)90256-8","article-title":"The Harry Dym equation in 2+1 dimensions: A reciprocal link with the Kadomtsev-Petviashvili equation","volume":"120","year":"1987","journal-title":"Phys. Lett. A"},{"key":"2023080108475977000_c29","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/0020-7462(94)90033-7","article-title":"An invariance of the (2+1)-dimensional Harry Dym equation: application to initial-boundary value problems","volume":"29","year":"1994","journal-title":"Int. J. Non-Linear Mech."},{"key":"2023080108475977000_c30","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/BF02750260","article-title":"On some spectral problems and isospectral evolutions connected with the classical string problem, I: Constants of motion","volume":"26","year":"1979","journal-title":"Lett. Nuovo Cimento Soc. Ital. Fis."},{"key":"2023080108475977000_c31","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1007\/BF02750261","article-title":"On some spectral problems and isospectral evolutions connected with the classical string problem, II: Evolution equation","volume":"26","year":"1979","journal-title":"Lett. Nuovo Cimento Soc. Ital. Fis."},{"first-page":"85","volume-title":"Nonlinear Evolution Equations and Dynamical Systems","year":"1979","key":"2023080108475977000_c32"},{"key":"2023080108475977000_c33","doi-asserted-by":"publisher","first-page":"226","DOI":"10.1007\/BF01075696","article-title":"A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem","volume":"8","year":"1974","journal-title":"Funct. Anal. Appl."}],"container-title":["Journal of Mathematical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/doi\/10.1063\/1.2982232\/15827401\/092901_1_online.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jmp\/article-pdf\/doi\/10.1063\/1.2982232\/15827401\/092901_1_online.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,1]],"date-time":"2023-08-01T08:49:32Z","timestamp":1690879772000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jmp\/article\/49\/9\/092901\/911713\/Bi-Hamiltonian-aspects-of-a-matrix-Harry-Dym"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,9,1]]},"references-count":33,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2008,9,1]]}},"URL":"https:\/\/doi.org\/10.1063\/1.2982232","relation":{},"ISSN":["0022-2488","1089-7658"],"issn-type":[{"type":"print","value":"0022-2488"},{"type":"electronic","value":"1089-7658"}],"subject":[],"published-other":{"date-parts":[[2008,9]]},"published":{"date-parts":[[2008,9,1]]},"article-number":"092901"}}