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Meromorphic Painlevé III transcendents and the Joukowski correspondence 亚纯painlevevl3超验与Joukowski对应
Journal of Integrable Systems Pub Date : 2018-09-20 DOI: 10.1093/INTEGR/XYZ001
Andrea E. V. Ferrari, L. Mason
{"title":"Meromorphic Painlevé III transcendents and the Joukowski correspondence","authors":"Andrea E. V. Ferrari, L. Mason","doi":"10.1093/INTEGR/XYZ001","DOIUrl":"https://doi.org/10.1093/INTEGR/XYZ001","url":null,"abstract":"We study a twistor correspondence based on the Joukowski map reduced from one for stationary-axisymmetric self-dual Yang-Mills and adapt it to the Painleve III equation. A natural condition on the geometry (axissimplicity) leads to solutions that are meromorphic at the fixed singularity at the origin. We show that it also implies a quantisation condition for the parameter in the equation. From the point of view of generalized monodromy data, the condition is equivalent to triviality of the Stokes matrices and half-integral exponents of formal monodromy. We obtain canonically defined representations in terms of a Birkhoff factorization whose entries are related to the data at the origin and the Painleve constants.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121177675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations 超几何微分方程的Voros系数和Eynard-Orantin的拓扑递推:第二部分:超几何方程的合流族
Journal of Integrable Systems Pub Date : 2018-05-28 DOI: 10.1093/INTEGR/XYZ004
Kohei Iwaki, T. Koike, Yumiko Takei
{"title":"Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion: Part II: For confluent family of hypergeometric equations","authors":"Kohei Iwaki, T. Koike, Yumiko Takei","doi":"10.1093/INTEGR/XYZ004","DOIUrl":"https://doi.org/10.1093/INTEGR/XYZ004","url":null,"abstract":"\u0000 We show that each member of the confluent family of the Gauss hypergeometric equations is realized as quantum curves for appropriate spectral curves. As an application, relations between the Voros coefficients of those equations and the free energy of their classical limit computed by the topological recursion are established. We will also find explicit expressions of the free energy and the Voros coefficients in terms of the Bernoulli numbers and Bernoulli polynomials.\u0000 Communicated by: Youjin Zhang","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127785146","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 22
Jeu de taquin, uniqueness of rectification and ultradiscrete KP Jeu de taquin，整流的唯一性和超离散KP
Journal of Integrable Systems Pub Date : 2018-03-03 DOI: 10.1093/integr/xyz012
Shinsuke Iwao
{"title":"Jeu de taquin, uniqueness of rectification and ultradiscrete KP","authors":"Shinsuke Iwao","doi":"10.1093/integr/xyz012","DOIUrl":"https://doi.org/10.1093/integr/xyz012","url":null,"abstract":"\u0000 In this article, we study tropical-theoretic aspects of the ‘rectification algorithm’ on skew Young tableaux. It is shown that the algorithm is interpreted as a time evolution of some tropical integrable system. By using this fact, we construct a new combinatorial map that is essentially equivalent to the rectification algorithm. Some of properties of the rectification can be seen more clearly via this map. For example, the uniqueness of a rectification boils down to an easy combinatorial problem. Our method is mainly based on the two previous researches: the theory of geometric tableaux by Noumi–Yamada, and the study on the relationship between jeu de taquin slides and the ultradiscrete KP equation by Mikami and Katayama–Kakei.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129856977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Higher solutions of Hitchin’s self-duality equations Hitchin自对偶方程的高解
Journal of Integrable Systems Pub Date : 2018-01-08 DOI: 10.1093/INTEGR/XYAA006
Lynn Heller, Sebastian Heller
{"title":"Higher solutions of Hitchin’s self-duality equations","authors":"Lynn Heller, Sebastian Heller","doi":"10.1093/INTEGR/XYAA006","DOIUrl":"https://doi.org/10.1093/INTEGR/XYAA006","url":null,"abstract":"Solutions of Hitchin’s self-duality equations correspond to special real sections of the Deligne–Hitchin moduli space—twistor lines. A question posed by Simpson in 1997 asks whether all real sections give rise to global solutions of the self-duality equations. An affirmative answer would in principle allow for complex analytic procedures to obtain all solutions of the self-duality equations. The purpose of this article is to construct counter examples given by certain (branched) Willmore surfaces in three-space (with monodromy) via the generalized Whitham flow. Though these sections do not give rise to global solutions of the self-duality equations on the whole Riemann surface M, they induce solutions on an open and dense subset of it. This suggest a connection between Willmore surfaces, i.e., rank 4 harmonic maps theory, with the rank 2 self-duality theory.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127528724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Discrete local induction equation 离散局部感应方程
Journal of Integrable Systems Pub Date : 2017-08-05 DOI: 10.1093/INTEGR/XYZ003
Sampei Hirose, J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta
{"title":"Discrete local induction equation","authors":"Sampei Hirose, J. Inoguchi, K. Kajiwara, N. Matsuura, Y. Ohta","doi":"10.1093/INTEGR/XYZ003","DOIUrl":"https://doi.org/10.1093/INTEGR/XYZ003","url":null,"abstract":"The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schr\"odinger equation. In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schr\"odinger equation. We also present explicit formulas for both smooth and discrete curves in terms of $tau$ functions of the two-component KP hierarchy.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132525202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
The algebraic Bethe Ansatz and combinatorial trees 代数Bethe Ansatz与组合树
Journal of Integrable Systems Pub Date : 2017-07-09 DOI: 10.1093/INTEGR/XYZ002
Ricardo Soares Vieira, A. Lima-Santos
{"title":"The algebraic Bethe Ansatz and combinatorial trees","authors":"Ricardo Soares Vieira, A. Lima-Santos","doi":"10.1093/INTEGR/XYZ002","DOIUrl":"https://doi.org/10.1093/INTEGR/XYZ002","url":null,"abstract":"We present in this paper a comprehensive introduction to the algebraic Bethe Ansatz, taking as examples the six-vertex model with periodic and non-periodic boundary conditions. We propose a diagrammatic representation of the commutation relations used in the algebraic Bethe Ansatz, so that the action of the transfer matrix in the nth excited state gives place to labeled combinatorial trees. The analysis of these combinatorial trees provides in a straightforward way the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations. Several identities between the R-matrix elements can also be derived from the symmetry of these diagrams regarding the permutation of their labels. This combinatorial approach gives some insights about how the algebraic Bethe Ansatz works, which can be valuable for non-experts readers.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"s3-23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130100488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Continuum limits of pluri-Lagrangian systems 多拉格朗日系统的连续统极限
Journal of Integrable Systems Pub Date : 2017-06-21 DOI: 10.1093/integr/xyy020
Mats Vermeeren
{"title":"Continuum limits of pluri-Lagrangian systems","authors":"Mats Vermeeren","doi":"10.1093/integr/xyy020","DOIUrl":"https://doi.org/10.1093/integr/xyy020","url":null,"abstract":"A pluri-Lagrangian (or Lagrangian multiform) structure is an attribute of integrability that has mainly been studied in the context of multidimensionally consistent lattice equations. It unifies multidimensional consistency with the variational character of the equations. An analogous continuous structure exists for integrable hierarchies of differential equations. We present a continuum limit procedure for pluri-Lagrangian systems. In this procedure the lattice parameters are interpreted as Miwa variables, describing a particular embedding in continuous multi-time of the mesh on which the discrete system lives. Then we seek differential equations whose solutions interpolate the embedded discrete solutions. The continuous systems found this way are hierarchies of differential equations. We show that this continuum limit can also be applied to the corresponding pluri-Lagrangian structures. We apply our method to the discrete Toda lattice and to equations H1 and Q1$_{delta = 0}$ from the ABS list.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123003986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 13
A Lagrangian View on Complete Integrability of the Two-Component Camassa–Holm System 双分量Camassa-Holm系统完全可积性的拉格朗日观点
Journal of Integrable Systems Pub Date : 2016-05-19 DOI: 10.1093/integr/xyx002
Jonathan Eckhardt, Katrin Grunert
{"title":"A Lagrangian View on Complete Integrability of the Two-Component Camassa–Holm System","authors":"Jonathan Eckhardt, Katrin Grunert","doi":"10.1093/integr/xyx002","DOIUrl":"https://doi.org/10.1093/integr/xyx002","url":null,"abstract":"We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa–Holm system can be understood in terms of certain reparametrizations of the underlying isospectral problem. The respective coordinates correspond to different normalizations of an associated first order system. In particular, we will see that the two-component Camassa–Holm system in Lagrangian variables is completely integrable as well.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130017476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
A q-deformation of discrete dynamical systems associated with the Weyl group of type A 与A型Weyl群相关的离散动力系统的q-变形
Journal of Integrable Systems Pub Date : 2015-09-01 DOI: 10.1093/INTEGR/XYW004
Azusa Ikeda, T. Masuda
{"title":"A q-deformation of discrete dynamical systems associated with the Weyl group of type A","authors":"Azusa Ikeda, T. Masuda","doi":"10.1093/INTEGR/XYW004","DOIUrl":"https://doi.org/10.1093/INTEGR/XYW004","url":null,"abstract":"A $q$-deformation of discrete dynamical systems associated with the Weyl group of type $A$ is proposed. This is a natural generalization of the birational Weyl group action for the $q$-Painleve equation of type $A_4^{(1)}$. A determinant formula of Jacobi–Trudi type for a family of Laurent polynomials arising from the action of the Weyl group is also presented.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132794030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Geometric characterization and classification of Bäcklund transformations of sine-Gordon type 正弦戈登型Bäcklund变换的几何表征与分类
Journal of Integrable Systems Pub Date : 2013-11-16 DOI: 10.1093/INTEGR/XYY018
J. Clelland, T. Ivey
{"title":"Geometric characterization and classification of Bäcklund transformations of sine-Gordon type","authors":"J. Clelland, T. Ivey","doi":"10.1093/INTEGR/XYY018","DOIUrl":"https://doi.org/10.1093/INTEGR/XYY018","url":null,"abstract":"We begin by considering several properties commonly (but not universally) possessed by Backlund transformations between hyperbolic Monge-Ampere equations: wavelike nature of the underlying equations, preservation of independent variables, quasilinearity of the transformation, and autonomy of the transformation. We show that, while these properties all appear to depend on the formulation of both the underlying PDEs and the Backlund transformation in a particular coordinate system, in fact they all have intrinsic geometric meaning, independent of any particular choice of local coordinates. \u0000Next, we consider the problem of classifying Backlund transformations with these properties. We show that, apart from a family of transformations between Monge-integrable equations, there exists only a finite-dimensional family of such transformations, including the well-known family of Backlund transformations for the sine-Gordon equation. The full extent of this family is not yet determined, but our analysis has uncovered previously unknown transformations among generalizations of Liouville's equation.","PeriodicalId":242196,"journal":{"name":"Journal of Integrable Systems","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115031949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
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