Drinfeld Sokolov Research Articles

Tau function of the CKP hierarchy and nonlinearizable virasoro symmetries

We introduce a single tau function that represents the C-type Kadomtsev–Petviashvili (CKP) hierarchy in a generalized Hirota ‘bilinear’ equation. The actions on the tau function by additional symmetries for the hierarchy are also calculated, which involve strictly more than a central extension of the -algebra. As an application, for Drinfeld–Sokolov hierarchies associated to affine Kac–Moody algebras of type C, we obtain a formula to compute the obstacles in linearizing their Virasoro symmetries and hence prove the Virasoro symmetries to be nonlinearizable when acting on the tau function.

Double reductions/analysis of the Drinfeld–Sokolov–Wilson equation

In this paper, we show that the recently developed formulation of the association between symmetries and conservation laws lead to double reductions of the third-order Drinfeld–Sokolov–Wilson system of partial differential equations (PDEs). For illustrative purposes only, we carry out the procedure for a version of the well-known third-order scalar Korteweg–de Vries (KdV) equation via a scaling symmetry of the equation.

From additional symmetries to linearization of Virasoro symmetries

We construct the additional symmetries and derive the Adler–Shiota–van Moerbeke formula for the two-component BKP hierarchy. Considered as certain reductions of the two-component BKP hierarchy, the Drinfeld–Sokolov hierarchies of type D are proved to possess symmetries written as the linear action of a series of Virasoro operators on the tau function. It results in that the Drinfeld–Sokolov hierarchies of type D coincide with Dubrovin and Zhang’s hierarchies associated to the Frobenius manifolds for Coxeter groups of type D, and that every solution of such a hierarchy together with the string equation is annihilated by certain combinations of the Virasoro operators and the time derivations of the hierarchy.

Auto-Bäcklund transformations and superposition formulas for solutions of Drinfeld–Sokolov systems

The paper is devoted to constructing Auto-Bäcklund transformations (ABT) and superposition formulas for the solutions of the Drinfeld–Sokolov (DS) systems. The transformations are derived from pairs of differential substitutions relating different systems of the DS type. The nonlinear superposition formulas for solutions of the DS systems are obtained from the assumption of commutativity of the Bianchi diagram. We indicate a seed solution for each system which can be used to generate multi-soliton solutions. As an application of the superposition formulas we construct two-soliton solutions for each of the DS systems.

Solving Toda field theories and related algebraic and differential properties

Toda field theories are important integrable systems. They can be regarded as constrained WZNW models, and this viewpoint helps to give their explicit general solutions, especially when a Drinfeld–Sokolov gauge is used. The main objective of this paper is to carry out this approach of solving the Toda field theories for the classical Lie algebras, following Balog et al. (1990) [5]. In this process, we discover and prove some algebraic identities for principal minors of special matrices. The known elegant solutions of Leznov (1980) [10] fit in our scheme in the sense that they are the general solutions to our conditions discovered in this solving process. To prove this, we find and prove some differential identities for iterated integrals. It can be said that altogether our paper gives complete mathematical proofs for Leznov’s solutions.

Traveling wave solutions [formula omitted] to a generalized Drinfel’d–Sokolov system which satisfy [formula omitted

An analysis of the coupled generalized Drinfel’d–Sokolov equationsut+α1uux+β1uxxx+γvδx=0andvt+α2uvx+β2vxxx=0is performed in the case of traveling wave solutions u and v satisfying the condition u=a1vm+a0. We are able to classify all such solutions in terms of the model parameters, and we then discuss the planar dynamics of such solutions. Numerical solutions are obtained and discussed for a variety of parameter regimes. Such results are one possible generalization of those given by Wu et al. [L. Wu, S. Chen, C. Pang, Traveling wave solutions for generalized Drinfeld–Sokolov equations, Applied Mathematical Modeling 33 (2009) 4126–4130].

A chiral Borel–Weil–Bott theorem

We compute the cohomology of modules over the algebra of twisted chiral differential operators over the flag manifold. This is applied to (1) finding the character of G-integrable irreducible highest weight modules over the affine Lie algebra at the critical level, and (2) computing a certain elliptic genus of the flag manifold. The main tool is a result that interprets the Drinfeld–Sokolov reduction as a derived functor.

Exp and modified Exp function methods for nonlinear Drinfeld–Sokolov system

In this paper, Exp-function and its modification methods have been applied to obtain an exact solution of the nonlinear Drinfeld–Sokolov system (DS). Modification of the method was first introduced by the same authors. The prominent merit of this method is to facilitate the process of solving systems of partial differential equations. These methods are straightforward and concise by themselves; moreover, their applications are promising to obtain exact solutions of various partial differential equations. It is shown that the methods, with the help of symbolic computation, provide very effective and powerful mathematical tools for solving such systems.

-algebras and surface operators in gauge theories

A general class of -algebras can be constructed from the affine sl(N) algebra by (quantum) Drinfeld–Sokolov reduction and are classified by partitions of N. Surface operators in an SU(N) 4D gauge theory are also classified by partitions of N. We argue that instanton partition functions of gauge theories in the presence of a surface operator can also be computed from the corresponding -algebra. We test this proposal by analysing the Polyakov–Bershadsky algebra obtaining results that are in agreement with the known partition functions for SU(3) gauge theories with a so-called simple surface operator. As a byproduct, our proposal implies relations between and algebras.

Drinfeld–Sokolov hierarchies on truncated current Lie algebras

Drinfeld–Sokolov hierarchies on truncated current Lie algebras

On the Drinfeld-Sokolov Hierarchies of D Type

We extend the notion of pseudo-differential operators that are used to represent the Gelfand–Dickey hierarchies and obtain a similar representation for the full Drinfeld–Sokolov hierarchies of D n type. By using such pseudo-differential operators, we introduce the tau functions of these bihamiltonian hierarchies and prove that these hierarchies are equivalent to the integrable hierarchies defined by Date–Jimbo–Kashiware–Miwa and Kac–Wakimoto from the basic representation of the Kac–Moody algebra D n (1) .

Soliton solutions of a few nonlinear wave equations

This paper carries out the integration of a few nonlinear wave equations to obtain topological as well as non-topological soliton solutions. The mathematical techniques used to obtain the soliton solutions are He’s variational iteration method, the tanh method and the ansatz method. The nonlinear wave equations that are studied are coupled mKdV equations, Drinfeld–Sokolov equation and its generalized version. Finally, some numerical simulations are given to support the analytical solutions.

LANGLANDS DUALITY IN LIOUVILLE-$H^3_+$ WZNW CORRESPONDENCE

We show a physical realization of the Langlands duality in correlation functions of [Formula: see text] WZNW model. We derive a dual version of the Stoyanovky–Riabult–Teschner (SRT) formula that relates the correlation function of the [Formula: see text] WZNW and the dual Liouville theory to investigate the level duality k - 2 → (k - 2)-1 in the WZNW correlation functions. Then, we show that such a dual version of the [Formula: see text]-Liouville relation can be interpreted as a particular case of a biparametric family of nonrational conformal field theories (CFT's) based on the Liouville correlation functions, which was recently proposed by Ribault. We study symmetries of these new nonrational CFT's and compute correlation functions explicitly by using the free field realization to see how a generalized Langlands duality manifests itself in this framework. Finally, we suggest an interpretation of the SRT formula as realizing the Drinfeld–Sokolov Hamiltonian reduction. Again, the Hamiltonian reduction reveals the Langlands duality in the [Formula: see text] WZNW model. Our new identity for the correlation functions of [Formula: see text] WZNW model may yield a first step to understand quantum geometric Langlands correspondence yet to be formulated mathematically.

Conservation laws for a complexly coupled KdV system, coupled Burgers’ system and Drinfeld–Sokolov–Wilson system via multiplier approach

The multiplier approach (variational derivative method) is used to derive the conservation laws for some nonlinear systems of partial differential equations. Firstly, the multipliers (characteristics) are computed and then conserved vectors are obtained for the each multiplier. Examples of the third-order complexly coupled KdV system, second-order coupled Burgers’ system and third-order Drinfeld–Sokolov–Wilson system are considered. For all three systems the local conservation laws are established by utilizing the multiplier approach.

KZ equation, G-opers, quantum Drinfeld–Sokolov reduction, and quantum Cayley–Hamilton identity

The Lax operator of Gaudin-type models is a 1-form at the classical level. In virtue of the quantization scheme proposed by D. Talalaev, it is natural to treat the quantum Lax operator as a connection; this connection is a partcular case of the Knizhnik–Zamolodchikov connection. In this paper, we find a gauge trasformation that produces the “second normal form,” or the “Drinfeld–Sokolov” form. Moreover, the differential operator nurally corresponding to this form is given precisely by the quantum characteristic polynomial of the Lax operator (this operator is called the G-oper or Baxter operator). This observation allows us to relate solutions of the KZ and Baxter equations in an obvious way, and to prove that the immanent KZ equation has only meromorphic solutions. As a corollary, we obtain the quantum Cayley–Hamilton identity for Gaudin-type Lax operators (including the general $ \mathfrak{gl}_{n}[t] $ case). The presented construction sheds a new light on the geometric Langlands correspondence. We also discuss the relation with the Harish-Chandra homomorphism. Bibliography: 19 titles.

Traveling wave solutions for Generalized Drinfeld–Sokolov equations

In this paper, solitary waves and periodic waves for Generalized Drinfeld–Sokolov equations are studied, by using the theory of dynamical systems. Bifurcation parameter sets are shown. Under given parameter conditions, explicit formulas of solitary wave, kink (anti-kink) wave and periodic wave solutions are obtained.

The Cole–Hopf transformation and multiple soliton solutions for the integrable sixth-order Drinfeld–Sokolov–Satsuma–Hirota equation

The completely integrable sixth-order nonlinear Drinfeld–Sokolov–Satsuma–Hirota equation is studied. Three distinct methods, namely the Cole–Hopf transformation method, the tanh–coth method, and the Exp-function method are used for a reliable treatment of this equation. Solitons, multiple soliton solutions, multiple singular soliton solutions, and plane periodic solutions are obtained for this equations. The study highlights the power of each method.

Frobenius manifolds and central invariants for the Drinfeld–Sokolov bihamiltonian structures

The Drinfeld–Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete sets of invariants of the related bihamiltonian structures with respect to the group of Miura-type transformations.

On classification and construction of algebraic Frobenius manifolds

We develop the theory of generalized bi-Hamiltonian reduction. Applying this theory to a suitable loop algebra we recover a generalized Drinfeld–Sokolov reduction. This gives a way to construct new examples of algebraic Frobenius manifolds.

Exact and numerical solutions of generalized Drinfeld–Sokolov equations

In this Letter, we consider a system of generalized Drinfeld–Sokolov (gDS) equations which models one-dimensional nonlinear wave processes in two-component media. We find some exact solutions of gDS by using tanh function method and we also obtain a numerical solution by using the Adomian's Decomposition Method (ADM).