scholarly journals THE LAX OPERATOR APPROACH FOR THE VIRASORO AND THE W-CONSTRAINTS IN THE GENERALIZED KdV HIERARCHY

1993 ◽  
Vol 08 (20) ◽  
pp. 3457-3478 ◽  
Author(s):  
SUDHAKAR PANDA ◽  
SHIBAJI ROY
Keyword(s):  
Symmetry Operator ◽  
String Equation ◽  
Kp Hierarchy ◽  
Operator Approach ◽  
Additional Symmetry ◽  
Kdv Hierarchy ◽  
Lax Operator

We show directly in the Lax operator approach how the Virasoro and W-constraints on the τ-function arise in the p-reduced KP hierarchy or generalized KdV hierarchy. In particular, we consider the KdV and the Boussinesq hierarchy to show that the Virasoro and the W-constraints follow from the string equation by expanding the "additional symmetry" operator in terms of the Lax operator. We also mention how this method could be generalized for higher KdV hierarchies.

Sato–Bäcklund transformations and string equations of the mKP hierarchy

2019 ◽  
Vol 34 (25) ◽  
pp. 1950142 ◽  
Author(s):  
Huizhan Chen ◽  
Lumin Geng ◽  
Jipeng Cheng
Keyword(s):  
Bäcklund Transformations ◽  
Kp Hierarchy ◽  
Time And Space ◽  
Additional Symmetry ◽  
Tau Functions ◽  
Virasoro Constraint ◽  
Backlund Transformations ◽  
Lax Operator ◽  
Important Kind

Additional symmetry is an important kind of symmetries depending explicitly on the time and space variables, which can be expressed through Sato–Bäcklund transformations. In this paper, we construct Sato–Bäcklund transformations of the modified KP hierarchy and its constrained cases. Then the string equations of the [Formula: see text]-reduced modified KP hierarchy are established by requiring the system independent on some additional symmetry flows, which are expressed by the Lax operator [Formula: see text] and the Orlov–Shulman’s operator [Formula: see text]. At last, we obtain the negative Virasoro constraint on the two tau functions of the 2-reduced modified KP hierarchy satisfying the string equations.

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.

Gauge Transformation for BC r -KP Hierarchy and Its Compatibility with Additional Symmetry

2019 ◽  
Vol 71 (3) ◽  
pp. 274 ◽  
Author(s):  
Lu-Min Geng ◽  
Hui-Zhan Chen ◽  
Na Li ◽  
Ji-Peng Cheng
Keyword(s):  
Gauge Transformation ◽  
Kp Hierarchy ◽  
Additional Symmetry

scholarly journals The Hamiltonian structures of the super‐KP hierarchy associated with an even parity super‐Lax operator

1995 ◽  
Vol 36 (1) ◽  
pp. 258-267 ◽  
Author(s):  
J. Barcelos‐Neto ◽  
Sasanka Ghosh ◽  
Shibaji Roy
Keyword(s):  
Kp Hierarchy ◽  
Hamiltonian Structures ◽  
Lax Operator

TETRACRITICAL ISING MODEL COUPLED TO THE TWO-DIMENSIONAL GRAVITY

1992 ◽  
Vol 07 (19) ◽  
pp. 4487-4499
Author(s):  
ALOK KUMAR ◽  
JNANADEVA MAHARANA
Keyword(s):  
Ising Model ◽  
Central Charge ◽  
Continuum Theory ◽  
Two Dimensional ◽  
Operator Approach ◽  
Modular Invariant ◽  
Lax Operator ◽  
Dimensional Gravity ◽  
Ade Classification ◽  
The Continuum

Nonperturbative string equations are found explicitly for a central charge c=4/5 model coupled to the two-dimensional quantum gravity in the Lax operator approach proposed by Douglas. These string equations are used to derive the scaling behavior of several correlation functions on the sphere and it is shown that they agree with the calculations of the continuum theory. The model, identified with the diagonal modular invariant in the ADE classification, corresponds to the tetracritical Ising model.

The reduction of the q-Wronskian solutions of the q-deformed modified KP hierarchy to its constrained case

2021 ◽  
pp. 2150295
Author(s):  
Ge Yi ◽  
Wei Wang ◽  
Kelei Tian ◽  
Ying Xu
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Kp Hierarchy ◽  
Lax Operator ◽  
Symmetry Constraints ◽  
Necessary And Sufficient

By using the eigenfunction symmetry constraints of the [Formula: see text]-deformed modified Kadomtsev–Petviashvili ([Formula: see text]-mKP) hierarchy, we show a necessary and sufficient condition to reduce [Formula: see text]-Wronskian solutions of the [Formula: see text]-mKP hierarchy to the [Formula: see text]-cmKP ([Formula: see text]-deformed constrained modified Kadomtsev–Petviashvili) hierarchy defined by setting the Lax operator as [Formula: see text]. Then an illustrative example is given.

scholarly journals On additional symmetry: the many-body problem related to the KP hierarchy

1994 ◽  
Vol 191 (1-2) ◽  
pp. 87-90 ◽  
Author(s):  
Kazuhiro Hikami ◽  
Miki Wadati
Keyword(s):  
Body Problem ◽  
Many Body ◽  
Kp Hierarchy ◽  
Additional Symmetry ◽  
The Many
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