scholarly journals DISPERSIONLESS BIGRADED TODA HIERARCHY AND ITS ADDITIONAL SYMMETRY

2012 ◽  
Vol 24 (07) ◽  
pp. 1230003 ◽  
Author(s):  
CHUANZHONG LI ◽  
JINGSONG HE
Keyword(s):  
Additional Symmetry ◽  
Tau Function ◽  
Toda Hierarchy ◽  
Definition Of ◽  
Sato Theory ◽  
Bilinear Equations

In this paper, we firstly give the definition of dispersionless bigraded Toda hierarchy (dBTH) and introduce some Sato theory on dBTH. Then we define Orlov–Schulman's [Formula: see text], [Formula: see text] operator and give the additional Block symmetry of dBTH. Meanwhile we give tau function of dBTH and some related dispersionless bilinear equations.

scholarly journals Tau function and Hirota bilinear equations for the extended bigraded Toda hierarchy

2010 ◽  
Vol 51 (4) ◽  
pp. 043514 ◽  
Author(s):  
Chuanzhong Li ◽  
Jingsong He ◽  
Ke Wu ◽  
Yi Cheng
Keyword(s):  
Tau Function ◽  
Toda Hierarchy ◽  
Hirota Bilinear ◽  
Bilinear Equations

scholarly journals Intertwining operator and integrable hierarchies from topological strings

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Jean-Emile Bourgine
Keyword(s):  
Topological Strings ◽  
Vertex Operator ◽  
Integrable Hierarchies ◽  
Kp Hierarchy ◽  
Tau Function ◽  
Time Deformation ◽  
Toroidal Algebra ◽  
Crystal Model ◽  
Melting Crystal ◽  
Definition Of

Abstract In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum W1+∞ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner — or vertex operator — of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $$ \mathfrak{gl} $$ gl (1) (also called Ding-Iohara-Miki algebra). In this way, we take a first step toward the definition of deformed hierarchies associated to A-model refined topological strings.

The q-deformed mKP hierarchy with two parameters

2018 ◽  
Vol 32 (16) ◽  
pp. 1850170
Author(s):  
Kelei Tian ◽  
Yanyan Ge ◽  
Xiaoming Zhu
Keyword(s):  
Additional Symmetry ◽  
Lax Equation ◽  
Quantum Calculus ◽  
Two Parameters ◽  
Sato Theory

In this paper, with the help of the biparametric quantum calculus we construct the Sato theory on the q-deformation modified Kadomtsev–Petviashvili hierarchy with two parameters (qp-mKP), which is a new deformation of classical mKP hierarchy. The Lax equation and dressing operator of qp-mKP hierarchy are derived. By considering the M operator and [Formula: see text] operator, the additional symmetry of qp-mKP hierarchy is obtained.

Symmetries of the multi-component Boussinesq hierarchy

2021 ◽  
pp. 2150111
Author(s):  
Meiyan Hu ◽  
Chuanzhong Li
Keyword(s):  
Wave Function ◽  
Generating Function ◽  
String Equation ◽  
Additional Symmetry ◽  
Infinite Dimensional ◽  
Lax Operator ◽  
Sato Theory ◽  
The Way

In this paper, we construct the Lax operator of the multi-component Boussinesq hierarchy. Based on the Sato theory and the dressing structure of the multi-component Boussinesq hierarchy, the adjoint wave function and the Orlov–Schulman’s operator are introduced, which are useful for constructing the additional symmetry of the multi-component Boussinesq hierarchy. Besides, the additional flows can commute with the original flows, and these flows form an infinite dimensional [Formula: see text] algebra. Taking the above discussion into account, we mainly study the additional symmetry flows and the generating function for both strongly and weakly multi-component of the Boussinesq hierarchies. By the way, using the [Formula: see text] constraint of the multi-component Boussinesq hierarchy, the string equation can be derived.

scholarly journals FAY-LIKE IDENTITIES OF THE TODA LATTICE HIERARCHY AND ITS DISPERSIONLESS LIMIT

2006 ◽  
Vol 18 (10) ◽  
pp. 1055-1073 ◽  
Author(s):  
LEE-PENG TEO
Keyword(s):  
Toda Lattice ◽  
Tau Function ◽  
Toda Hierarchy ◽  
Dispersionless Limit ◽  
Toda Lattice Hierarchy ◽  
Hirota Equations

In this paper, we derive the Fay-like identities of tau function for the Toda lattice hierarchy from the bilinear identity. We prove that the Fay-like identities are equivalent to the hierarchy. We also show that the dispersionless limit of the Fay-like identities are the dispersionless Hirota equations of the dispersionless Toda hierarchy.

scholarly journals Multifold Darboux Transformations of the Extended Bigraded Toda Hierarchy

2016 ◽  
Vol 71 (4) ◽  
pp. 357-371 ◽  
Author(s):  
Chuanzhong Li ◽  
Tao Song
Keyword(s):  
Soliton Solutions ◽  
Vertex Operators ◽  
Darboux Transformations ◽  
Flow Equations ◽  
Toda Hierarchy ◽  
Witten Theory ◽  
Approximation Analysis ◽  
The Difference ◽  
Hirota Bilinear ◽  
Bilinear Equations

AbstractWith the extended logarithmic flow equations and some extended Vertex operators in generalized Hirota bilinear equations, extended bigraded Toda hierarchy (EBTH) was proved to govern the Gromov-Witten theory of orbiford cNM in literature. The generating function of these Gromov-Witten invariants is one special solution of the EBTH. In this article, the multifold Darboux transformations and their determinant representations of the EBTH are given with two different gauge transformation operators. The two Darboux transformations in different directions are used to generate new solutions from known solutions which include soliton solutions of (N, N)-EBTH, i.e. the EBTH when N=M. From the generation of new solutions, we can find the big difference between the EBTH and the extended Toda hierarchy (ETH). Also, we plotted the soliton graphs of the (N, N)-EBTH from which some approximation analysis is given. From the analysis on velocities of soliton solutions, the difference between the extended flows and other flows are shown. The two different Darboux transformations constructed by us might be useful in Gromov-Witten theory of orbiford cNM.

Introductory paper

1966 ◽  
Vol 24 ◽  
pp. 3-5
Author(s):  
W. W. Morgan
Keyword(s):  
Line Intensity ◽  
Classification Scheme ◽  
Two Dimensional ◽  
Spectral Classification ◽  
Dimensional Array ◽  
Rr Lyrae ◽  
Metallic Line ◽  
Introductory Paper ◽  
Parameter Varying ◽  
Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

2.2. Working Group 2: Conceptual Definitions of Reference Coordinate Systems for Earth Dynamics

1975 ◽  
Vol 26 ◽  
pp. 21-26
Keyword(s):  
Coordinate System ◽  
Working Group ◽  
General Character ◽  
Coordinate Systems ◽  
Reference Coordinate System ◽  
Group 2 ◽  
Definition Of ◽  
Earth Dynamics

An ideal definition of a reference coordinate system should meet the following general requirements:1. It should be as conceptually simple as possible, so its philosophy is well understood by the users.2. It should imply as few physical assumptions as possible. Wherever they are necessary, such assumptions should be of a very general character and, in particular, they should not be dependent upon astronomical and geophysical detailed theories.3. It should suggest a materialization that is dynamically stable and is accessible to observations with the required accuracy.

Symbiotic Stars at Optical, Infrared and Radio Wavelengths

1979 ◽  
Vol 46 ◽  
pp. 125-149 ◽  
Author(s):  
David A. Allen
Keyword(s):  
Temperature Regime ◽  
Type Star ◽  
Stellar Systems ◽  
Symbiotic Stars ◽  
Lower Excitation ◽  
Definition Of ◽  
Original Type

No paper of this nature should begin without a definition of symbiotic stars. It was Paul Merrill who, borrowing on his botanical background, coined the termsymbioticto describe apparently single stellar systems which combine the TiO absorption of M giants (temperature regime ≲ 3500 K) with He II emission (temperature regime ≳ 100,000 K). He and Milton Humason had in 1932 first drawn attention to three such stars: AX Per, CI Cyg and RW Hya. At the conclusion of the Mount Wilson Ha emission survey nearly a dozen had been identified, and Z And had become their type star. The numbers slowly grew, as much because the definition widened to include lower-excitation specimens as because new examples of the original type were found. In 1970 Wackerling listed 30; this was the last compendium of symbiotic stars published.

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