{"title":"参数Korteweg-de Vries层次和超椭圆Sigma函数","authors":"E. Yu. Bunkova, V. M. Bukhshtaber","doi":"10.1134/S0016266322030029","DOIUrl":null,"url":null,"abstract":"<p> In this paper, a parametric Korteweg–de Vries hierarchy is defined that depends on an infinite set of graded parameters <span>\\(a = (a_4,a_6,\\dots)\\)</span>. It is shown that, for any genus <span>\\(g\\)</span>, the Klein hyperelliptic function <span>\\(\\wp_{1,1}(t,\\lambda)\\)</span> defined on the basis of the multidimensional sigma function <span>\\(\\sigma(t, \\lambda)\\)</span>, where <span>\\(t = (t_1, t_3,\\dots, t_{2g-1})\\)</span> and <span>\\(\\lambda = (\\lambda_4, \\lambda_6,\\dots, \\lambda_{4 g + 2})\\)</span>, specifies a solution to this hierarchy in which the parameters <span>\\(a\\)</span> are given as polynomials in the parameters <span>\\(\\lambda\\)</span> of the sigma function. The proof uses results concerning the family of operators introduced by V. M. Buchstaber and S. Yu. Shorina. This family consists of <span>\\(g\\)</span> third-order differential operators in <span>\\(g\\)</span> variables. Such families are defined for all <span>\\(g \\geqslant 1\\)</span>, the operators in each of them pairwise commute with each other and also commute with the Schrödinger operator. In this paper a relationship between these families and the Korteweg–de Vries parametric hierarchy is described. A similar infinite family of third-order operators on an infinite set of variables is constructed. The results obtained are extended to the case of such a family. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"56 3","pages":"169 - 187"},"PeriodicalIF":0.6000,"publicationDate":"2023-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Parametric Korteweg–de Vries Hierarchy and Hyperelliptic Sigma Functions\",\"authors\":\"E. Yu. Bunkova, V. M. Bukhshtaber\",\"doi\":\"10.1134/S0016266322030029\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p> In this paper, a parametric Korteweg–de Vries hierarchy is defined that depends on an infinite set of graded parameters <span>\\\\(a = (a_4,a_6,\\\\dots)\\\\)</span>. It is shown that, for any genus <span>\\\\(g\\\\)</span>, the Klein hyperelliptic function <span>\\\\(\\\\wp_{1,1}(t,\\\\lambda)\\\\)</span> defined on the basis of the multidimensional sigma function <span>\\\\(\\\\sigma(t, \\\\lambda)\\\\)</span>, where <span>\\\\(t = (t_1, t_3,\\\\dots, t_{2g-1})\\\\)</span> and <span>\\\\(\\\\lambda = (\\\\lambda_4, \\\\lambda_6,\\\\dots, \\\\lambda_{4 g + 2})\\\\)</span>, specifies a solution to this hierarchy in which the parameters <span>\\\\(a\\\\)</span> are given as polynomials in the parameters <span>\\\\(\\\\lambda\\\\)</span> of the sigma function. The proof uses results concerning the family of operators introduced by V. M. Buchstaber and S. Yu. Shorina. This family consists of <span>\\\\(g\\\\)</span> third-order differential operators in <span>\\\\(g\\\\)</span> variables. Such families are defined for all <span>\\\\(g \\\\geqslant 1\\\\)</span>, the operators in each of them pairwise commute with each other and also commute with the Schrödinger operator. In this paper a relationship between these families and the Korteweg–de Vries parametric hierarchy is described. A similar infinite family of third-order operators on an infinite set of variables is constructed. The results obtained are extended to the case of such a family. </p>\",\"PeriodicalId\":575,\"journal\":{\"name\":\"Functional Analysis and Its Applications\",\"volume\":\"56 3\",\"pages\":\"169 - 187\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2023-01-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Functional Analysis and Its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S0016266322030029\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Functional Analysis and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S0016266322030029","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Parametric Korteweg–de Vries Hierarchy and Hyperelliptic Sigma Functions
In this paper, a parametric Korteweg–de Vries hierarchy is defined that depends on an infinite set of graded parameters \(a = (a_4,a_6,\dots)\). It is shown that, for any genus \(g\), the Klein hyperelliptic function \(\wp_{1,1}(t,\lambda)\) defined on the basis of the multidimensional sigma function \(\sigma(t, \lambda)\), where \(t = (t_1, t_3,\dots, t_{2g-1})\) and \(\lambda = (\lambda_4, \lambda_6,\dots, \lambda_{4 g + 2})\), specifies a solution to this hierarchy in which the parameters \(a\) are given as polynomials in the parameters \(\lambda\) of the sigma function. The proof uses results concerning the family of operators introduced by V. M. Buchstaber and S. Yu. Shorina. This family consists of \(g\) third-order differential operators in \(g\) variables. Such families are defined for all \(g \geqslant 1\), the operators in each of them pairwise commute with each other and also commute with the Schrödinger operator. In this paper a relationship between these families and the Korteweg–de Vries parametric hierarchy is described. A similar infinite family of third-order operators on an infinite set of variables is constructed. The results obtained are extended to the case of such a family.
期刊介绍:
Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.