scholarly journals Differential Fay identities and auxiliary linear problem of integrable hierarchies

2019 ◽  
Author(s):  
Kanehisa Takasaki
Keyword(s):  
Linear Problem ◽  
Integrable Hierarchies ◽  
Auxiliary Linear Problem

scholarly journals Discrete Hirota's Equation in Quantum Integrable Models

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3125-3158 ◽  
Author(s):  
A. Zabrodin
Keyword(s):  
Recent Progress ◽  
Toda Lattice ◽  
Linear Problem ◽  
Classical Equation ◽  
Transfer Matrices ◽  
Hirota Equation ◽  
Quantum Integrable Models ◽  
Classical Integrable ◽  
Specific Quantum ◽  
Auxiliary Linear Problem

The recent progress in revealing classical integrable structures in quantum models solved by Bethe ansatz is reviewed. Fusion relations for eigenvalues of quantum transfer matrices can be written in the form of classical Hirota's bilinear difference equation. This equation is also known as the completely discretized version of the 2D Toda lattice. We explain how one obtains the specific quantum results by solving the classical equation. The auxiliary linear problem for the Hirota equation is shown to generalize Baxter's T-Q relation.

scholarly journals Coupled Nonlinear Dynamics in the Three-Mode Integrable System on a Regular Chain

2021 ◽  
Vol 66 (7) ◽  
pp. 601
Author(s):  
O.O. Vakhnenko
Keyword(s):  
Linear Problem ◽  
Hamiltonian Formulation ◽  
Dynamical Equations ◽  
Third Order ◽  
Nonlinear Dynamical ◽  
One Dimensional ◽  
Zero Curvature Representation ◽  
Coupled Subsystems ◽  
Regular Chain ◽  
Auxiliary Linear Problem

The article suggests the nonlinear lattice system of three dynamical subsystems coupled both in their potential and kinetic parts. Due to its essentially multicomponent structure the system is capable to model nonlinear dynamical excitations on regular quasi-one-dimensional lattices of various physical origins. The system admits a clear Hamiltonian formulation with the standard Poisson structure. The alternative Lagrangian formulation of system’s dynamics is also presented. The set of dynamical equations is integrable in the Lax sense, inasmuch as it possesses a zero-curvature representation. Though the relevant auxiliary linear problem involves a spectral third-order operator, we have managed to develop an appropriate two-fold Darboux–Backlund dressing technique allowing one to generate the nontrivial crop solution embracing all three coupled subsystems in a rather unusual way.

scholarly journals Duality of positive and negative integrable hierarchies via relativistically invariant fields

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
S. Y. Lou ◽  
X. B. Hu ◽  
Q. P. Liu
Keyword(s):  
Integrable Hierarchies ◽  
Gordon Equation ◽  
Formal Series ◽  
Two Dimensional ◽  
Relativistic Invariance ◽  
Sine Gordon Equation ◽  
Recursion Operators ◽  
Symmetry Approach ◽  
Duality Relations ◽  
Duality Problem

Abstract It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

Loop algebra and visaoro symmetries of integrable hierarchies of KP type

2001 ◽  
Vol 78 (3-4) ◽  
pp. 233-253 ◽  
Author(s):  
H. Aratyn ◽  
J.F. Gomes ◽  
E. Nissimov ◽  
S. Pacheva
Keyword(s):  
Integrable Hierarchies ◽  
Loop Algebra

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.

scholarly journals Cooperative Operation Schedules of Energy Storage System and Demand Response Resources Considering Urban Railway Load Characteristic under a Time-of-Use Tariff

2021 ◽  
Vol 16 (3) ◽  
pp. 1273-1284
Author(s):  
Hye Ji Kim ◽  
Hosung Jung ◽  
Young Jun Ko ◽  
Eun Su Chae ◽  
Hyo Jin Kim ◽  
...  
Keyword(s):  
Energy Storage ◽  
Demand Response ◽  
Air Conditioning ◽  
Linear Problem ◽  
Storage System ◽  
Energy Charge ◽  
Energy Storage System ◽  
Peak Reduction ◽  
Railway Stations ◽  
Charging And Discharging Processes

AbstractThis paper proposes an algorithm for the cooperative operation of air conditioning facilities and the energy storage system (ESS) in railway stations to minimize electricity. Unlike traditional load patterns, load patterns of an urban railway station can peak where energy charge rates are not high. Due to this possibility, if applying the traditional peak-reduction algorithm to railway loads, energy changes can increase, resulting in higher electricity bills. Therefore, it is required to develop a new method for minimizing the sum of capacity charges and energy charges, which is a non-linear problem. To get a feasible solution for this problem, we suggest an algorithm that optimizes the facility operation through two optimizations (primary and secondary). This method is applied to the air-quality change model for operating air conditioning facilities as demand-response (DR) resources in railway stations. This algorithm makes it possible to estimate operable DR capacity every hour, rather than calculating the capacity of DR resources conservatively in advance. Finally, we perform a simulation for the application of the proposed method to the operation of DR resources and ESS together. The simulation shows that electricity bills become lowered, and the number of charging and discharging processes of ESS is also reduced.

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