Additional symmetries of the Two-Boson hierarchy and the multi-component Two-Boson hierarchy

2020 ◽  
pp. 2150073
Author(s):  
Jian Li ◽  
Tiecheng Xia
Keyword(s):  
Lie Algebra ◽  
Flow Equations ◽  
Additional Symmetries ◽  
Basic Facts ◽  
Time Variables

In this paper, we firstly recall some basic facts on the Two-Boson hierarchy. Then, introducing some time variables that consist of a non-abelian Lie algebra. Next, we construct additional symmetries for the Two-Boson hierarchy with the aid of the Orlov–Schulman operator, which depend on the time variables and dressing operator. In addition, we give the additional flow equations of the Two-Boson hierarchy as a simple example, and prove that the additional flows are symmetries of the Two-Boson hierarchy. In this way, an isomorphism between the additional symmetries of the Two-Boson hierarchy and the [Formula: see text] algebra is constructed. Finally, the multi-component Two-Boson hierarchy can be defined, and we consider the additional symmetries for the multi-component Two-Boson hierarchy with the method of Dickey, Orlov, and Shulman.

Additional symmetries for the supersymmetric Gelfand–Dickey hierarchy

2020 ◽  
Vol 17 (11) ◽  
pp. 2050164
Author(s):  
Jian Li ◽  
Chuanzhong Li
Keyword(s):  
Lie Superalgebra ◽  
Type Condition ◽  
Flow Equations ◽  
Text Type ◽  
Additional Symmetries ◽  
Time Variables

In this paper, we first define a supersymmetric Gelfand–Dickey hierarchy using the Sato equation. Then we introduce some time variables and derivations involving those that from a non-abelian Lie superalgebra. Next, we construct additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the aid of the Orlov–Schulman operators which involve the above time variables and dressing operators. We give the additional flow equations of the supersymmetric Gelfand–Dickey hierarchy, and prove that the additional flows are symmetries of the supersymmetric Gelfand–Dickey hierarchy. In this way, an isomorphism between the additional symmetries of the supersymmetric Gelfand–Dickey hierarchy and the super [Formula: see text] algebra is constructed. Finally, the [Formula: see text] type supersymmetric Gelfand–Dickey hierarchy is constructed and we consider the additional symmetries for the supersymmetric Gelfand–Dickey hierarchy with the [Formula: see text] type condition which is compatible with the flows of the [Formula: see text] type supersymmetric Gelfand–Dickey hierarchy.

On sojourn time in Jackson networks of queues

1987 ◽  
Vol 24 (02) ◽  
pp. 495-510 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Sojourn Time ◽  
Network Flow ◽  
External Input ◽  
Single Server ◽  
Jackson Networks ◽  
Flow Equations ◽  
Markovian Queue ◽  
The Laplace Transform ◽  
Time Variables ◽  
Networks Of Queues

This paper is about representations for equilibrium sojourn time distributions in Jackson networks of queues. For a network with N single-server nodes let hi be the Laplace transform of the residual system sojourn time for a customer ‘arriving' to node i, ‘arrival' meaning external input or internal transfer. The transforms {hi : i = 1, ···, N} are shown to satisfy a system of equations we call the network flow equations. These equations lead to a general recursive representation for the higher moments of the sojourn time variables {Ti : i = 1, ···, N}. This recursion is discussed and then, by way of illustration, applied to the single-server Markovian queue with feedback.

scholarly journals P∞ Algebra of KP, Free Fermions and 2-Cocycle in the Lie Algebra of Pseudodifferential Operators

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3159-3193 ◽  
Author(s):  
A. Yu. Orlov ◽  
P. Winternitz
Keyword(s):  
Lie Algebra ◽  
Pseudodifferential Operators ◽  
Virasoro Algebra ◽  
Symmetry Algebra ◽  
Lie Point Symmetries ◽  
Negative Part ◽  
Fermion Algebra ◽  
Kp Equations ◽  
Time Variables ◽  
Positive Half

The symmetry algebra P∞=W∞⊕ H ⊕ I∞ of integrable systems is defined. As an example the classical Sophus Lie point symmetries of all higher KP equations are obtained. It is shown that one ("positive") half of the point symmetries belongs to the W∞ symmetries while the other ("negative") part belongs to the I∞ ones. The corresponding action on the τ-function is obtained. A new embedding of the Virasoro algebra into gl(∞) describes conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.

On sojourn time in Jackson networks of queues

1987 ◽  
Vol 24 (2) ◽  
pp. 495-510 ◽  
Author(s):  
Austin J. Lemoine
Keyword(s):  
Sojourn Time ◽  
Network Flow ◽  
External Input ◽  
Single Server ◽  
Jackson Networks ◽  
Flow Equations ◽  
Markovian Queue ◽  
The Laplace Transform ◽  
Time Variables ◽  
Networks Of Queues

This paper is about representations for equilibrium sojourn time distributions in Jackson networks of queues. For a network with N single-server nodes let hi be the Laplace transform of the residual system sojourn time for a customer ‘arriving' to node i, ‘arrival' meaning external input or internal transfer. The transforms {hi : i = 1, ···, N} are shown to satisfy a system of equations we call the network flow equations. These equations lead to a general recursive representation for the higher moments of the sojourn time variables {Ti : i = 1, ···, N}. This recursion is discussed and then, by way of illustration, applied to the single-server Markovian queue with feedback.

Dilation-free solutions for the incompressible flow equations on nonstaggered grids

AIAA Journal ◽  
1997 ◽  
Vol 35 ◽  
pp. 585-586
Author(s):  
P. A. Russell ◽  
S. Abdallah
Keyword(s):  
Incompressible Flow ◽  
Flow Equations

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

Sign in / Sign up

Export Citation Format

Share Document