scholarly journals Generalized q-Painlevé VI Systems of Type (A2n+1+A1+A1)(1) Arising From Cluster Algebra

2020 ◽  
Author(s):  
Naoto Okubo ◽  
Takao Suzuki
Keyword(s):  
Weyl Group ◽  
Cluster Algebra ◽  
Higher Order ◽  
Root Lattice ◽  
Similarity Reduction ◽  
Birational Transformations ◽  
System A ◽  
Affine Weyl Group ◽  
Painlevé Vi Equation ◽  
Garnier System

Abstract In this article we formulate a group of birational transformations that is isomorphic to an extended affine Weyl group of type $(A_{2n+1}+A_1+A_1)^{(1)}$ with the aid of mutations and permutations of vertices to a mutation-periodic quiver on a torus. This group provides a class of higher order generalizations of Jimbo–Sakai’s $q$-Painlevé VI equation as translations on a root lattice. Then the known three systems are obtained again: the $q$-Garnier system, a similarity reduction of the lattice $q$-UC hierarchy, and a similarity reduction of the $q$-Drinfeld–Sokolov hierarchy.

scholarly journals Symmetries in the fourth Painlevé equation and Okamoto polynomials

1999 ◽  
Vol 153 ◽  
pp. 53-86 ◽  
Author(s):  
Masatoshi Noumi ◽  
Yasuhiko Yamada
Keyword(s):  
Weyl Group ◽  
Schur Functions ◽  
Painlevé Equation ◽  
Symmetric Form ◽  
Rational Solutions ◽  
Kp Hierarchy ◽  
Similarity Reduction ◽  
Affine Weyl Group ◽  
Dependent Variables ◽  
Painleve Equation

AbstractThe fourth Painlevé equation PIV is known to have symmetry of the affine Weyl group of type with respect to the Bäcklund transformations. We introduce a new representation of PIV, called the symmetric form, by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of PIV is given in terms of this representation. Through the symmetric form, it turns out that PIV is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions PIV, called Okamoto polynomials, are expressible in terms of the 3-reduced Schur functions.

Resonant dynamics of an autoparametric system: A study using higher-order averaging

1996 ◽  
Vol 31 (1) ◽  
pp. 21-39 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj ◽  
Patricia Davies
Keyword(s):  
Higher Order ◽  
System A ◽  
Resonant Dynamics

The affine Weyl group and modular invariant partition functions

1988 ◽  
Vol 205 (2-3) ◽  
pp. 281-284 ◽  
Author(s):  
D. Altschüler ◽  
J. Lacki ◽  
Ph. Zaugg
Keyword(s):  
Weyl Group ◽  
Partition Functions ◽  
Modular Invariant ◽  
Affine Weyl Group

Square-root-like higher-order topological states in three-dimensional sonic crystals

2021 ◽  
Author(s):  
Zhiguo Geng ◽  
Huanzhao Lv ◽  
Zhan Xiong ◽  
Yu-Gui Peng ◽  
Zhaojiang Chen ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Surface States ◽  
Higher Order ◽  
Square Root ◽  
Root Lattice ◽  
Third Order ◽  
Topological Materials ◽  
Topological States ◽  
Sonic Crystal ◽  
Sonic Crystals

Abstract The square-root descendants of higher-order topological insulators were proposed recently, whose topological property is inherited from the squared Hamiltonian. Here we present a three-dimensional (3D) square-root-like sonic crystal by stacking the 2D square-root lattice in the normal (z) direction. With the nontrivial intralayer couplings, the opened degeneracy at the K-H direction induces the emergence of multiple acoustic localized modes, i.e., the extended 2D surface states and 1D hinge states, which originate from the square-root nature of the system. The square-root-like higher order topological states can be tunable and designed by optionally removing the cavities at the boundaries. We further propose a third-order topological corner state in the 3D sonic crystal by introducing the staggered interlayer couplings on each square-root layer, which leads to a nontrivial bulk polarization in the z direction. Our work sheds light on the high-dimensional square-root topological materials, and have the potentials in designing advanced functional devices with sound trapping and acoustic sensing.

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