Generalized Drinfeld Sokolov Research Articles

Virasoro constraints for Drinfeld–Sokolov hierarchies and equations of Painlevé type

We construct a tau cover of the generalized Drinfeld–Sokolov hierarchy associated with an arbitrary affine Kac–Moody algebra with gradations s ⩽ 1 $\mathrm{s}\leqslant \mathbb {1}$ and derive its Virasoro symmetries. By imposing the Virasoro constraints we obtain solutions of the Drinfeld–Sokolov hierarchy of Witten–Kontsevich and of Brezin–Gross–Witten types, and of those characterized by certain ordinary differential equations of Painlevé type. We also show the existence of affine Weyl group actions on the space of solutions of such ordinary differential equations, which generalizes the theory of Noumi and Yamada on affine Weyl group symmetries of the Painlevé-type equations.

Lie Symmetry Analysis, Conservation Laws, Power Series Solutions, and Convergence Analysis of Time Fractional Generalized Drinfeld-Sokolov Systems

In this work, we investigate invariance analysis, conservation laws, and exact power series solutions of time fractional generalized Drinfeld–Sokolov systems (GDSS) using Lie group analysis. Using Lie point symmetries and the Erdelyi–Kober (EK) fractional differential operator, the time fractional GDSS equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. Moreover, we have constructed conservation laws for time fractional GDSS and obtained explicit power series solutions of the reduced nonlinear ODEs that converge. Lastly, some figures are presented for explicit solutions.

The time fractional D(m,n) system: invariant analysis, explicit solution, conservation laws and optical soliton

ABSTRACT In this article, a generalized Drinfeld–Sokolov system, also called system which describes the anomalous propagation of nonlinear surface gravity waves over horizontal seabed, is investigated. The Lie classical method is performed on the governing model. The obtained symmetries are used to reduce the model into the fractional ordinary differential equations and to derive the local conservation laws. To make the use of the reduced system, an explicit series solution is derived. Also, it is shown that the bright and singular solitons exist only for the time fractional equation by means of a transformation. The considered system does not give rise to the dark soliton.

Poisson pencils: Reduction, exactness, and invariants

We study the invariants (in particular, the central invariants) of suitable Poisson pencils from the point of view of the theory of bi-Hamiltonian reduction, paying a particular attention to the case where the Poisson pencil is exact. We show that the exactness is preserved by the reduction. In the Drinfeld–Sokolov case, the same is true for the characteristic polynomial of the pencil, which plays a crucial role in the definition of the central invariants. We also discuss the bi-Hamiltonian structures of a generalized Drinfeld–Sokolov hierarchy and of the Camassa–Holm equation.

Classical Affine W-Superalgebras via Generalized Drinfeld–Sokolov Reductions and Related Integrable Systems

The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel'd-Sokolov (D-S) reduction associated to a Lie superalgebra $g$ and its even nilpotent element $f$, and we find a new definition of the classical affine W-superalgebra $W(g,f,k)$ via the D-S reduction. This new construction allows us to find free generators of $W(g,f,k)$, as a differential superalgebra, and two independent Lie brackets on $W(g,f,k)/\partial W(g,f,k).$ Moreover, we describe super-Hamiltonian systems with the Poisson vertex algebras theory. A W-superalgebra with certain properties can be understood as an underlying differential superalgebra of a series of integrable super-Hamiltonian systems.

Lie Point Symmetries and Travelling Wave Solutions for the Generalized Drinfeld–Sokolov System

ABSTRACTIn this article, we show successful methods with a wider applicability in physics, which have been applied to make an analysis of generalized Drinfeld–Sokolov system. Lie’s classical symmetries have been calculated and they have given us the Lie-group classification depending on the parameters. Moreover, for each case we have obtained their optimal system and reductions to conclude obtaining traveling wave solutions using the sine–cosine method.

Erratum to: Classical $${\mathcal {W}}$$ W -Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

Erratum to: Classical $${\mathcal {W}}$$ W -Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

Dirac Reduction for Poisson Vertex Algebras

We construct an analogue of Dirac’s reduction for an arbitrary local or non-local Poisson bracket in the general setup of non-local Poisson vertex algebras. This leads to Dirac’s reduction of an arbitrary non-local Poisson structure. We apply this construction to an example of a generalized Drinfeld–Sokolov hierarchy.

Classical $${\mathcal{W}}$$ W -Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

We derive explicit formulas for lambda-brackets of the affine classical W-algebras attached to the minimal and short nilpotent elements of any simple Lie algebra g. This is used to compute explicitly the first non-trivial PDE of the corresponding intgerable generalized Drinfeld-Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov's equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h-3 functions, where h is the dual Coxeter number of g. In the case when g is sl_2 both these equations coincide with the KdV equation. In the case when g is not of type C_n, we associate to the minimal nilpotent element of g yet another generalized Drinfeld-Sokolov hierarchy.

New applications of reduced differential transform method

In this paper, we applied the reduced differential transform method (RDTM) for solving two types of nonlinear partial differential equations, namely, generalized Drinfeld–Sokolov (gDS) equations and Kaup–Kupershmidt (KK) equation. Comparing the RDTM results with exact solutions shows that the RDTM is quite accurate, effective, reliable and can be applied for many other nonlinear partial differential equations.

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].

Trigonometric and hyperbolic type solutions to a generalized Drinfel’d–Sokolov equation

A class of trigonometric and hyperbolic type solutions to the generalized Drinfel’d–Sokolov (GDS) equations u t + α 1 uu x + β 1 u xxx + γ ( v δ ) x = 0 and v t + α 2 uv x + β 2 v xxx = 0 is obtained for the case in which α 2 = 0, for various values of the other model parameters. The method of homotopy analysis is then applied to obtain local analytical solutions for nonzero values of the parameter α 2, in effect extending the exact solutions. We do not assume traveling wave solution forms for the analytical solutions; that is, we solve the generalized Drinfel’d–Sokolov equations as PDEs without resorting to transforming the system to ODEs. An error analysis of the obtained approximate local analytical solutions is provided. Then, we outline a general framework by which one many construct solutions in either sine/cosine or sinh/cosh basis. We provide the general perturbation expansion via homotopy analysis, and we also discuss a method of selecting the convergence control parameter so as to minimize residual errors. Travelling solutions with time-dependent amplitude are then discussed.

Exponential-type solutions to a generalized Drinfel'd–Sokolov equation

Exact exponential-type solutions to the generalized Drinfel'd–Sokolov (GDS) equations are obtained for the case in which α2=0, for various values of the other model parameters. A modification of the homotopy analysis method is then applied to obtain analytical solutions for nonzero values of the parameter α2, in effect extending the exact solutions. In our modification of the standard method, we employ a nonlinear auxiliary operator. In contrast to most standard perturbation methods, in which a nonlinear problem is reduced to ‘infinitely many’ linear problems, here we reduce a hard nonlinear problem to ‘infinitely many’ easier nonlinear problems. Indeed, we also provide a solution using a linear auxiliary operator and show that the convergence of obtained solutions is improved (in the sense that fewer terms are required for the approximate solutions to obtain a desired accuracy) when using the auxiliary nonlinear operator, in some cases. An error analysis of the obtained approximate analytical solutions is provided.

Analytical solutions to a generalized Drinfel’d–Sokolov equation related to DSSH and KdV6

Analytical solutions to the generalized Drinfel’d–Sokolov (GDS) equations u t + α 1 uu x + β 1 u xxx + γ ( v δ ) x = 0 and v t + α 2 uv x + β 2 v xxx = 0 are obtained for various values of the model parameters. In particular, we provide perturbation solutions to illustrate the strong influence of the parameters β 1 and β 2 on the behavior of the solutions. We then consider a Miura-type transform which reduces the gDS equations into a sixth-order nonlinear differential equation under the assumption that δ = 1. Under such a transform the GDS reduces to the sixth-order Drinfel’d–Sokolov–Satsuma–Hirota (DSSH) equation (also known as KdV6) in the very special case α 1 = − α 2. The method of homotopy analysis is applied in order to obtain analytical solutions to the resulting equation for arbitrary α 1 and α 2. An error analysis of the obtained approximate analytical solutions is provided.

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.

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).

GAUGING OF GEOMETRIC ACTIONS AND INTEGRABLE HIERARCHIES OF THE KP TYPE

This work consists of two interrelated parts. First, we derive massive gauge-invariant generalizations of geometric actions on coadjoint orbits of arbitrary (infinite-dimensional) groups G with central extensions, with gauge group H being certain (infinite-dimensional) subgroup of G. We show that there exist generalized "zero-curvature" representations of the pertinent equations of motion on the coadjoint orbit. Second, in the special case of G being the Kac–Moody group, the equations of motion of the underlying gauged WZNW geometric action are identified as additional-symmetry flows of generalized Drinfeld–Sokolov integrable hierarchies based on the loop algebra [Formula: see text]. For [Formula: see text] the latter hierarchies are equivalent to the class c KP R,M of constrained (reduced) KP hierarchies. We describe in some detail the loop algebras of additional (nonisospectral) symmetries of c KP R,M hierarchies. Apart from gauged WZNW models, certain higher-dimensional nonlinear systems such as Davey–Stewartson and N-wave resonant systems are also identified as additional symmetry flows of c KP R,M hierarchies. Also we present the explicit derivation of the general Darboux–Bäcklund solutions of c KP R,M hierarchies preserving their additional (nonisospectral) symmetries, which for R=1 contain among themselves solutions to the gauged SL (M+1)/ U (1)× SL (M) WZNW field equations.

Constrained KP models as integrable matrix hierarchies

We formulate the constrained KP hierarchy (denoted by cKPK+1,M) as an affine sl̂(M+K+1) matrix integrable hierarchy generalizing the Drinfeld–Sokolov hierarchy. Using an algebraic approach, including the graded structure of the generalized Drinfeld–Sokolov hierarchy, we are able to find several new universal results valid for the cKP hierarchy. In particular, our method yields a closed expression for the second bracket obtained through Dirac reduction of any untwisted affine Kac–Moody current algebra. An explicit example is given for the case sl̂(M+K+1), for which a closed expression for the general recursion operator is also obtained. We show how isospectral flows are characterized and grouped according to the semisimple non-regular element E of sl(M+K+1) and the content of the center of the kernel of E.