{"id":115,"date":"2017-01-12T13:54:56","date_gmt":"2017-01-12T13:54:56","guid":{"rendered":"http:\/\/users.utu.fi\/hietarin\/?page_id=115"},"modified":"2017-08-01T13:29:03","modified_gmt":"2017-08-01T13:29:03","slug":"discrete","status":"publish","type":"page","link":"https:\/\/users.utu.fi\/hietarin\/home\/discrete\/","title":{"rendered":"Discrete Integrability"},"content":{"rendered":"<h6><strong>(This page is under construction and many items are still missing)<\/strong><\/h6>\n<h3><span style=\"color: #ff0000\">The Red Book<\/span><\/h3>\n<p><img decoding=\"async\" class=\"alignright\" src=\"https:\/\/assets.cambridge.org\/97811070\/42728\/cover\/9781107042728.jpg\" alt=\"Discrete Systems and Integrability\" \/><\/p>\n<p><strong>Discrete Systems and Integrability<\/strong> by J. Hietarinta, N. Joshi and F Nijhoff.<\/p>\n<p>&#8220;This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments.\u00a0 The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples.\u00a0 The book contains a thorough list of references.&#8221;<\/p>\n<p>Available at <a href=\"https:\/\/www.amazon.com\/Discrete-Systems-Integrability-Cambridge-Mathematics\/dp\/1107669480\/\">amazon.com<\/a>\u00a0 <a href=\"https:\/\/www.amazon.co.uk\/Discrete-Systems-Integrability-Cambridge-Mathematics\/dp\/1107669480\/\">amazon.co.uk<\/a> <a href=\"https:\/\/www.amazon.de\/Discrete-Systems-Integrability-Cambridge-Mathematics\/dp\/1107669480\/\">amazon.de<\/a> <a href=\"https:\/\/www.amazon.fr\/Discrete-Systems-Integrability-Cambridge-Mathematics\/dp\/1107669480\/\">amazon.fr<\/a> <a href=\"https:\/\/www.amazon.it\/Discrete-Systems-Integrability-Cambridge-Mathematics\/dp\/1107669480\/\">amazon.it<\/a>\u00a0 <a href=\"https:\/\/www.amazon.cn\/Discrete-Systems-Integrability-Cambridge-Mathematics\/dp\/1107669480\/\">amazon.cn\u00a0 <\/a><a href=\"http:\/\/www.amazon.in\/Discrete-Systems-Integrability-Cambridge-Mathematics\/dp\/1107669480\/\">amazon.in<\/a>\u00a0 <a href=\"https:\/\/www.amazon.co.jp\/Discrete-Systems-Integrability-Cambridge-Mathematics\/dp\/1107669480\/\">amazon.jp<\/a>\u00a0 and of course at <a href=\"http:\/\/www.cambridge.org\/gb\/academic\/subjects\/mathematics\/differential-and-integral-equations-dynamical-systems-and-co\/discrete-systems-and-integrabilit\">CUP .<\/a><\/p>\n<h3>Lectures and talks<\/h3>\n<p>(links still broken, also more talks to come)<\/p>\n<p>Talk at SIDE IX Bangalore 2014 and in Leeds 2015 on <a href=\"http:\/\/users.utu.fi\/hietarin\/leeds2015\/\"><strong>Discrete Boussinesq equations<\/strong>.<\/a><\/p>\n<p>Lectures series <b>Definitions and Predictions of Integrability for Difference Equations<\/b> given at the SMS-Summer School &#8220;Symmetries and Integrability of Difference Equations&#8221;, Montreal, June 9-21, 2008:<\/p>\n<ul>\n<li><a href=\"http:\/\/users.utu.fi\/hietarin\/home\/discrete\/sms1\/\">Lectures 1 and 2<\/a> (singularity confinement and algebraic entropy for maps)<\/li>\n<li><a href=\"sms2.pdf\">Lecture 3<\/a> (singularity confinement and algebraic entropy for lattices)<\/li>\n<li><a href=\"sms3.pdf\">Lecture 4<\/a> (consistency-around-the-cube)<\/li>\n<\/ul>\n<h3>Soliton solution to Lattice equations<\/h3>\n<ul>\n<li>F. Nijhoff,\u00a0 J Atkinson and J. Hietarinta: <span class=\"cite\" style=\"font-style: italic\"><strong class=\"cite_title\">Soliton solutions for ABS lattice equations: I. Cauchy matrix approach<\/strong><\/span><span style=\"font-style: italic\">,<\/span><a href=\"http:\/\/dx.doi.org\/10.1088\/1751-8113\/42\/40\/404005\"> J. Phys. A: Math. Theor. <b> 42<\/b> 404005 (34 pages) (2009).<\/a><\/li>\n<li>J. Hietarinta and Da-Jun Zhang: <span class=\"cite\" style=\"font-style: italic\"><strong class=\"cite_title\">Soliton solutions for ABS lattice equations: II. Casoratians and bilinearization<\/strong><\/span>,<a href=\"http:\/\/dx.doi.org\/10.1088\/1751-8113\/42\/40\/404006\"> J. Phys. A: Math. Theor. <b> 42<\/b> 404006 (30 pages) (2009).<\/a><\/li>\n<li>J Atkinson, J. Hietarinta and F Nijhoff: <em>Soliton solutions for Q3<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1088\/1751-8113\/41\/14\/142001\"> J. Phys. A: Math. Theor. <b> 41<\/b><br \/>\n142001 (11 pages) (2008).<\/a><\/li>\n<li>J Atkinson, J. Hietarinta and F Nijhoff: <em>Seed and soliton solutions for Adler&#8217;s lattice equation<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1088\/1751-8113\/40\/1\/F01\"> J. Phys. A: Math. Theor. <b> 40<\/b> F1-F8 (2007).<\/a><\/li>\n<\/ul>\n<h3>Searches<b><b> <\/b><\/b><\/h3>\n<ul>\n<li><b><b> J. Hietarinta and C. Viallet: <em>Searching for integrable lattice maps using factorization<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1088\/1751-8113\/40\/42\/S09\">J. Phys. A: Math. Theor. <b> 40<\/b> 12629-12643 (2007).<\/a><\/b><\/b><\/li>\n<li><b><b> J. Hietarinta: <em>Searching for CAC-maps<\/em>, <a href=\"http:\/\/dx.doi.org\/10.2991\/jnmp.2005.12.s2.16\">J. Nonlinear Math. Phys. <b>12<\/b> Suppl. 2, 223-230 (2005).<\/a> <\/b><\/b><\/li>\n<li><b><b> J. Hietarinta: <em>A new two-dimensional lattice model that is&#8217;consistent around the cube&#8217;<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1088\/0305-4470\/37\/6\/L01\">J. Phys. A: Math.<br \/>\nGen. <b>37<\/b>, L67-L73 (2004)<\/a><\/b><\/b><\/li>\n<\/ul>\n<h3>Singularity confinement and algebraic entropy<\/h3>\n<ul>\n<li><b><b> J. Hietarinta and C. Viallet: <em>Discrete Painlev\u00e9 I and singularity confinement in projective space<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1016\/S0960-0779%2898%2900266-5\">Chaos, Solitons and Fractals <b>11<\/b>, 29-32 (2000)<\/a>. <\/b><\/b><\/li>\n<li>J. Hietarinta and C. Viallet: <em>Singularity confinement and chaos in discrete systems<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1103\/PhysRevLett.81.325\">Phys. Rev. Lett.\u00a0 <b>81<\/b>, 325-328 (1998).<\/a><\/li>\n<li><b><b> J. Hietarinta: <em>Regularity of Difference Equations<\/em>, in <a href=\"http:\/\/www.worldscibooks.com\/mathematics\/5957.html\">&#8220;Difference Equations and Discrete Dynamical Systems&#8221;<\/a> eds. L. Allen, B. Aulbach, S. Elaydi and R. Sacker, (World Scientific, 2005) pp. 233-245. <\/b><\/b><\/li>\n<li><b><b> J. Hietarinta: <em>Integrability tests for difference equations<\/em>,\u00a0 <a href=\"http:\/\/dx.doi.org\/10.1016\/S0362-546X%2801%2900577-6\">Nonlinear Analysis <b>47<\/b>, 4641-4650 (2001)<\/a>. <\/b><\/b><\/li>\n<li><b><b> J. Hietarinta and C. Viallet: <em>Singularity Confinement and Degree Growth<\/em>, CRM Proceedings and Lecture Notes <b>25<\/b>,<a href=\"http:\/\/www.lpthe.jussieu.fr\/%7Eviallet\/hv_side3.ps.gz\"> 209-216\u00a0 <\/a>(2000). <\/b><\/b><\/li>\n<\/ul>\n<h3>Discrete Painlev\u00e9 equations<b><b> <\/b><\/b><\/h3>\n<ul>\n<li>A. Ramani, B. Grammaticos and J. Hietarinta: <em>Discrete versions of the Painlev\u00e9 Equation<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1103\/PhysRevLett.67.1829\">Phys. Rev. Lett. <b>67<\/b>, 1829-1831 (1991).<\/a><\/li>\n<li><b><b> J. Hietarinta and K. Kajiwara: <em>Rational solutions to d-P<sub>IV<\/sub> <\/em>, in <a href=\"http:\/\/www.cambridge.org\/catalogue\/catalogue.asp?isbn=9780521596992\">&#8220;Symmetries and Integrability of Difference Equations&#8221;<\/a>, eds. P. Clarkson and F. Nijhoff, London Mathematical Society Lecture Note Series 255\u00a0 (Cambridge UP, 1999) pp. 206-216. <\/b><\/b><\/li>\n<li><b><b>J. Satsuma, K. Kajiwara, B. Grammaticos, J. Hietarinta and A. Ramani: <em>Bilinear discrete Painlev\u00e9-II and its particular solutions<\/em>,\u00a0 <a href=\"http:\/\/dx.doi.org\/10.1088\/0305-4470\/28\/12\/025\">J. Phys. A: Math. Gen. <b> 28<\/b>, 3541-3548 (1995).<\/a> <\/b><\/b><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>(This page is under construction and many items are still missing) The Red Book Discrete Systems and Integrability by J. Hietarinta, N. Joshi and F Nijhoff. &#8220;This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. &hellip; <a href=\"https:\/\/users.utu.fi\/hietarin\/home\/discrete\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Discrete Integrability&#8221;<\/span><\/a><\/p>\n","protected":false},"author":178,"featured_media":0,"parent":43,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-115","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/115","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/users\/178"}],"replies":[{"embeddable":true,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/comments?post=115"}],"version-history":[{"count":10,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/115\/revisions"}],"predecessor-version":[{"id":1328,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/115\/revisions\/1328"}],"up":[{"embeddable":true,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/43"}],"wp:attachment":[{"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/media?parent=115"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}