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In this paper we introduce the notion of coalgebra symmetry for discrete systems. With this concept we prove that all discrete radially symmetric systems in standard form are quasi-integrable and that all variational discrete quasi-radially…

The generalized Drinfeld-Sokolov construction of KdV systems is reviewed in the case of an arbitrary affine Lie algebra paying particular attention to Hamiltonian aspects and $\W$-algebras. Some extensions of known results as well as a new…

We introduce a hierarchy of integrable PDEs in 2+1 dimensions arising from the commutation of 2-dimensional vector fields. We also solve the associated Cauchy problems, using the recently developed Inverse Scattering Transform for…

We represent an algorithm allowing one to construct new classes of partially integrable multidimensional nonlinear partial differential equations (PDEs) starting with the special type of solutions to the (1+1)-dimensional hierarchy of…

Weight systems are functions on chord diagrams satisfying Vassiliev's $4$-term relations. They originate in the theory of finite type knot invariants. Recent developments in understanding weight systems arising from Lie algebras are based…

For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $\pi$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},\pi)$-type. We then derive the Sato-Zhou type formula for tau functions of the…

The basic concepts underlying our analysis of {\it W-algebras} as extended symmetries of integrable systems are summarized. The construction starts from the second hamiltonian structure of ``Generalized Drinfel'd-Sokolov'' hierarchies, and…

In this paper, we construct the additional symmetries of the supersymmetric BKP(SBKP) hierarchy. These additional flows constitute a B type $SW_{1+\infty}$ Lie algebra because of the B type reduction of the supersymmetric BKP hierarchy.…

We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which…

Inspired by a recent work of Dubrovin [7], for each simple Lie algebra $\mathfrak{g}$, we introduce an infinite family of pairwise commuting ODEs and define their $\tau$-functions. We show that these $\tau$-functions can be identified with…

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…

Turaev conjectured that the classification, realization and splitting results for Poincar\'e duality complexes of dimension $3$ (PD$_{3}$-complexes) generalize to PD$_{n}$-complexes with $(n-2)$-connected universal cover for $n \ge 3$.…

We compute the first cohomology group of the symmetric algebra of the universal \'etale $p$-adic local system on the tower of coverings of Drinfeld's $p$-adic half-plane. The result takes a factorized form, using the $p$-adic Langlands…

In the previous work we introduced the higher order $q$-Painlev\'{e} system $q$-$P_{(n+1,n+1)}$ as a generalization of the Jimbo-Sakai's $q$-Painlev\'{e} VI equation. It is derived from a $q$-analogue of the Drinfeld-Sokolov hierarchy of…

Recently we showed how, in two-dimensional scalar theories, one-loop threshold diagrams can be cut into the product of one or more tree-level diagrams arXiv:2206.09368. Using this method on the ADE series of Toda models, we computed the…

The tau-function formalism for a class of generalized ``zero-curvature'' integrable hierarchies of partial differential equations, is constructed. The class includes the Drinfel'd-Sokolov hierarchies. A direct relation between the variables…

In this paper, the classification in [Lin,Wei,Ye] of solutions to Toda systems of type $A$ with singular sources is generalized to Toda systems of types $C$ and $B$. Like in the $A$ case, the solution space is shown to be parametrized by…

We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS$_{3}$ algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis…

We report on generalizations of the KdV-type integrable hierarchies of Drinfel'd and Sokolov. These hierarchies lead to the existence of new classical $W$-algebras, which arise as the second Hamiltonian structure of the hierarchies. In…

A new approach to integrability of affine Toda field theories and closely related to them KdV hierarchies is proposed. The flows of a hierarchy are explicitly identified with infinitesimal action of the principal abelian subalgebra of the…