{"id":49,"date":"2017-01-12T08:52:18","date_gmt":"2017-01-12T08:52:18","guid":{"rendered":"http:\/\/users.utu.fi\/hietarin\/?page_id=49"},"modified":"2017-07-27T08:07:48","modified_gmt":"2017-07-27T08:07:48","slug":"hirotas-bilinear-method","status":"publish","type":"page","link":"https:\/\/users.utu.fi\/hietarin\/home\/hirotas-bilinear-method\/","title":{"rendered":"Hirota&#8217;s bilinear method"},"content":{"rendered":"<h3>Introduction to Hirota&#8217;s bilinear method (in conference proceedings)<\/h3>\n<ul>\n<li>J. Hietarinta: <a href=\"https:\/\/users.utu.fi\/hietarin\/wp-content\/uploads\/sites\/169\/2017\/01\/Bangalore.pdf\">Hirota&#8217;s bilinear method and integrability&#8221;<\/a> (Bangalore, 18-29 February, 2008)<\/li>\n<li>J.Hietarinta: <em>Introduction to the bilinear method<\/em>, in &#8220;<a href=\"http:\/\/www.springer.com\/physics\/book\/978-3-540-20630-9\">Integrability of Nonlinear Systems<\/a>&#8221;, eds. Y. Kosman-Schwarzbach, B. Grammaticos and K.M. Tamizhmani, Lecture Notes in Physics Vol. 638 (Springer, 2004) pp. 95-103.<\/li>\n<li>J.Hietarinta: <em> B\u00e4cklund transformations from the bilinear viewpoint<\/em>,<a href=\"http:\/\/www.ams.org\/bookstore?fn=20&amp;arg1=crmpseries&amp;item=CRMP-29\"> CRM Proceedings and Lecture Notes <b>29<\/b> 245-251 (2001).<\/a><\/li>\n<li>J.Hietarinta: <em> Hirota&#8217;s bilinear method and its generalization<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1142\/S0217751X97000062\">Int. J. Mod. Phys. A <b>12<\/b>, 43-51 (1997).<\/a><\/li>\n<li>J.Hietarinta: <em>Hirota&#8217;s bilinear method and partial integrability<\/em>, in <a href=\"http:\/\/www.springer.com\/physics\/book\/978-0-7923-0752-5\">&#8220;Partially Integrable Equations in Physics&#8221;<\/a>, eds. R. Conte and N. Boccara, (Kluwer, 1990), pp. 459-478.<\/li>\n<\/ul>\n<h3>Soliton solutions<\/h3>\n<ul>\n<li>C. Gilson, J. Hietarinta, J. Nimmo and Y. Ohta: <em>Sasa-Satsuma higher-order nonlinear Schr\u00f6dinger equation<br \/>\nand its bilinearization and multisoliton solutions<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1103\/PhysRevE.68.016614\">Phys. Rev. E <b>68<\/b>, 016614 (2003)<\/a><\/li>\n<li>J. Hietarinta: <em>Comment on `Exact solutions of the two-dimensional Burgers equation&#8217;<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1088\/0305-4470\/33\/28\/401\"> J. Phys. A:<br \/>\nMath. Gen. <b> 33<\/b> 5157-5158 (2000).<\/a><\/li>\n<li>R. Radhakrishnan, M. Lakshmanan and J. Hietarinta: <em>Inelastic collision and switching of coupled bright solitons in optical fibers<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1103\/PhysRevE.56.2213\">Phys. Rev. E\u00a0 <b>56<\/b>, 2213-2216 (1997).<\/a><\/li>\n<li>J. Hietarinta, A. Ramani and B. Grammaticos: <em>Continuous vacua in bilinear soliton equations<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1088\/0305-4470\/27\/9\/027\"> J. Phys. A:<br \/>\nMath. Gen. <b> 27<\/b>, 3149-3158 (1994).\u00a0<\/a><\/li>\n<\/ul>\n<h3>Dromions<\/h3>\n<ul>\n<li>J. Hietarinta: <em>One-dromion solutions for generic classes of equations<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1016\/0375-9601(90)90535-V\"> Phys. Lett. A <b>149<\/b>, 113-118 (1990).<\/a><\/li>\n<li>J. Hietarinta and R. Hirota: <em>Multidromion solutions to the Davey-Stewartson equation<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1016\/0375-9601(90)90357-T\"> Phys. Lett. A <b>145<\/b>, 237-244 (1990).<\/a><\/li>\n<li>J.Hietarinta: <em> On the dromion masses<\/em>, in &#8220;Nonlinear Evolution Equations and Dynamical Systems, Proceedings NEEDS&#8217;91&#8221;, eds. M.\u00a0 Boiti, P. Martina and F. Pempinelli, (World Scientific, 1992), pp. 23-29.<\/li>\n<li>J.Hietarinta: <em> From an analytical formula to a movie by way of REDUCE and C<\/em>, in &#8220;Proceedings of the Workshop on Symbolic and Numeric Computation&#8221;, eds. H. Apiola, M. Laine and E. Valkeila, (Computing Centre, University of Helsinki, 1991) Research Reports N:o 17, pp. 117-126.<\/li>\n<li>J.Hietarinta: <em> 2+1 Dimensional Dromions and Hirota&#8217;s Bilinear Method<\/em>, in &#8220;Solitons and Chaos&#8221;, eds. I. Antoniou and F.J. Lambert, (Springer, 1991), pp. 321-324.<\/li>\n<li>J.Hietarinta: <em> Dromion Solutions for Generic NLS- and KdV-Type Equations<\/em>, in &#8220;Nonlinear Evolution Equations and Dynamical Systems, NEEDS&#8217;90&#8221;, eds. V.G. Makhankov and O.K. Pashaev, (Springer,<br \/>\n1991), pp. 83-85.<\/li>\n<\/ul>\n<h3>The three-soliton condition as a search method<\/h3>\n<ul>\n<li>B. Grammaticos, A. Ramani and J. Hietarinta: <em>A search for integrable bilinear equations: the Painlev\u00e9 approach<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1063\/1.529005\">J. Math. Phys. <b>31<\/b>, 2572-2578 (1990).<\/a><\/li>\n<li>J. Hietarinta: <em>A search of bilinear equations passing Hirota&#8217;s three-soliton condition: IV. Complex bilinear equations<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1063\/1.528002\">J. Math. Phys. <b>29<\/b>, 628-635 (1988).<\/a><\/li>\n<li>J. Hietarinta: <em>A search of bilinear equations passing Hirota&#8217;s three-soliton condition: III. sine-Gordon-type bilinear equations<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1063\/1.527750\">J. Math. Phys. <b>28<\/b>, 2586-2592 (1987).<\/a><\/li>\n<li>J. Hietarinta: <em>A search of bilinear equations passing Hirota&#8217;s three-soliton condition: II. mKdV-type bilinear<br \/>\nequations<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1063\/1.527421\"> J. Math. Phys. <b>28<\/b>, 2094-2101 (1987).<\/a><\/li>\n<li>J. Hietarinta: <em>A search of bilinear equations passing Hirota&#8217;s three-soliton condition: I. KdV-type bilinear equations<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1063\/1.527815\">J. Math. Phys. <b>28<\/b>, 1732-1742 (1987).<\/a><\/li>\n<li>J.Hietarinta: <em> Searching for integrable PDE&#8217;s by testing Hirota&#8217;s three-soliton condition<\/em>, in &#8220;Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISSAC&#8217;91&#8221;, ed. Stephen M. Watt, (Association for Computing Machinery, 1991),<a href=\"http:\/\/doi.acm.org\/10.1145\/120694.120739\"> pp. 295-300.<\/a><\/li>\n<li>J.Hietarinta: <em> Equations That Pass Hirota&#8217;s Three-Soliton Condition and Other Tests of Integrability<\/em>, in &#8220;Nonlinear Evolution Equations and Dynamical Systems&#8221;, eds. S. Carillo and O. Ragnisco, (Springer,<br \/>\n1990), pp. 46-50.<\/li>\n<li>J.Hietarinta: <em> Recent results from the search for bilinear equations having three-soliton solutions<\/em>, in &#8220;Nonlinear evolution equations: integrability and spectral methods&#8221;, eds. A. Degasperis, A.P. Fordy and M. Lakshmanan, (Manchester U.P., 1990) pp. 307-317.<\/li>\n<\/ul>\n<h3>Trilinear equations<\/h3>\n<ul>\n<li>J.Hietarinta: <em> Gauge symmetry and the generalization of Hirota&#8217;s bilinear method<\/em>, <a href=\"http:\/\/dx.doi.org\/10.2991\/jnmp.1996.3.3-4.2\">J. Nonlin. Math. Phys. <b>3<\/b>, 260-265 (1996).<\/a><\/li>\n<li>B. Grammaticos, A. Ramani and J. Hietarinta: <em>Multilinear operators: the natural extension of Hirota&#8217;s bilinear formalism<\/em>,<a href=\"http:\/\/dx.doi.org\/10.1016\/0375-9601(94)90367-0\"> Phys. Lett. A <b>190<\/b>, 65-70 (1994).\u00a0<\/a><\/li>\n<li>J. Satsuma, K. Kajiwara, J. Matsukidaira and J. Hietarinta: <em>Solution to the Broer-Kaup system through its trilinear form<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1143\/JPSJ.61.3096\">J. Phys. Soc. Jpn. <b>61<\/b>, 3096-3102 (1992).<\/a><\/li>\n<li>J. Hietarinta and J. Satsuma: <em>The trilinear equation as a 2+2 dimensional extension of the 1+1 dimensional relativistic Toda lattice<\/em>, <a href=\"http:\/\/dx.doi.org\/10.1016\/0375-9601(91)90015-Z\">Phys. Lett. A <b>161<\/b>, 267-273 (1991).<\/a><\/li>\n<li>J.Hietarinta, B. Grammaticos and A. Ramani: <em>Integrable Trilinear PDE&#8217;s<\/em>, in &#8220;Nonlinear Evolution Equations &amp; Dynamical Systems, NEEDS &#8217;94&#8221;, eds. V.G. Makhankov, A.R. Bishop and D.D. Holm, (World Scientific, 1995), pp. 54-63.<\/li>\n<li>J.Hietarinta, K. Kajiwara, J. Matsukidaira and J. Satsuma: <em> The Relativistic Toda Lattice and Its Trilinear Form<\/em>, in &#8220;Nonlinear Evolution Equations and Dynamical Systems, Proceedings NEEDS&#8217;91&#8221;, eds. M. Boiti, P. Martina and F. Pempinelli, (World Scientific, 1992), pp. 30-43.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Hirota&#8217;s bilinear method (in conference proceedings) J. Hietarinta: Hirota&#8217;s bilinear method and integrability&#8221; (Bangalore, 18-29 February, 2008) J.Hietarinta: Introduction to the bilinear method, in &#8220;Integrability of Nonlinear Systems&#8221;, eds. Y. Kosman-Schwarzbach, B. Grammaticos and K.M. Tamizhmani, Lecture Notes in Physics Vol. 638 (Springer, 2004) pp. 95-103. J.Hietarinta: B\u00e4cklund transformations from the bilinear viewpoint, &hellip; <a href=\"https:\/\/users.utu.fi\/hietarin\/home\/hirotas-bilinear-method\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Hirota&#8217;s bilinear method&#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-49","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/49","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=49"}],"version-history":[{"count":6,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/49\/revisions"}],"predecessor-version":[{"id":1229,"href":"https:\/\/users.utu.fi\/hietarin\/wp-json\/wp\/v2\/pages\/49\/revisions\/1229"}],"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=49"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}