{"title":"论波戈雅夫伦斯基网格的三浦变换与带算子的逆谱问题之间的联系","authors":"Andrey Osipov","doi":"10.52737/18291163-2024.16.2-1-28","DOIUrl":null,"url":null,"abstract":"We consider semi-infinite and finite Bogoyavlensky lattices$$\\overset\\cdot a_i =a_i\\left(\\prod_{j=1}^{p}a_{i+j}-\\prod_{j=1}^{p}a_{i-j}\\right),$$$$\\overset\\cdot b_i = b_i\\left(\\sum_{j=1}^{p}b_{i+j}-\\sum_{j=1}^{p}b_{i-j}\\right),$$for some $p\\ge 1$, and Miura-like transformations between these systems, defined for $p\\ge 2$. Both lattices are integrable (via Lax pair formalism) by the inverse spectral problem method for band operators, i.e., operators generated by band matrices. The key role in this method is played by the moments of the Weyl matrix of the corresponding band operator and their evolution in time. We find a description of the above-mentioned transformations in terms of these moments and apply this result to study finite Bogoyavlensky lattices and, in particular, their first integrals.","PeriodicalId":42323,"journal":{"name":"Armenian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Links between Miura Transformations of Bogoyavlensky Lattices and Inverse Spectral Problems for Band Operators\",\"authors\":\"Andrey Osipov\",\"doi\":\"10.52737/18291163-2024.16.2-1-28\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider semi-infinite and finite Bogoyavlensky lattices$$\\\\overset\\\\cdot a_i =a_i\\\\left(\\\\prod_{j=1}^{p}a_{i+j}-\\\\prod_{j=1}^{p}a_{i-j}\\\\right),$$$$\\\\overset\\\\cdot b_i = b_i\\\\left(\\\\sum_{j=1}^{p}b_{i+j}-\\\\sum_{j=1}^{p}b_{i-j}\\\\right),$$for some $p\\\\ge 1$, and Miura-like transformations between these systems, defined for $p\\\\ge 2$. Both lattices are integrable (via Lax pair formalism) by the inverse spectral problem method for band operators, i.e., operators generated by band matrices. The key role in this method is played by the moments of the Weyl matrix of the corresponding band operator and their evolution in time. We find a description of the above-mentioned transformations in terms of these moments and apply this result to study finite Bogoyavlensky lattices and, in particular, their first integrals.\",\"PeriodicalId\":42323,\"journal\":{\"name\":\"Armenian Journal of Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-02-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Armenian Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.52737/18291163-2024.16.2-1-28\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Armenian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.52737/18291163-2024.16.2-1-28","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the Links between Miura Transformations of Bogoyavlensky Lattices and Inverse Spectral Problems for Band Operators
We consider semi-infinite and finite Bogoyavlensky lattices$$\overset\cdot a_i =a_i\left(\prod_{j=1}^{p}a_{i+j}-\prod_{j=1}^{p}a_{i-j}\right),$$$$\overset\cdot b_i = b_i\left(\sum_{j=1}^{p}b_{i+j}-\sum_{j=1}^{p}b_{i-j}\right),$$for some $p\ge 1$, and Miura-like transformations between these systems, defined for $p\ge 2$. Both lattices are integrable (via Lax pair formalism) by the inverse spectral problem method for band operators, i.e., operators generated by band matrices. The key role in this method is played by the moments of the Weyl matrix of the corresponding band operator and their evolution in time. We find a description of the above-mentioned transformations in terms of these moments and apply this result to study finite Bogoyavlensky lattices and, in particular, their first integrals.