Related papers: Category of nonlinear evolution equations, algebra…
The definition of index for differential algebraic equations (DAEs) or integral algebraic equations (IAEs) in the linear case (time variable) depends only on the coefficients of integrals or differential operators and the coefficients of…
It is sometimes desirable to produce for a nonlinear system of ODEs a new representation of simpler structural form, but it is well known that this goal may imply an increase in the dimension of the system. This is what happens if in this…
For a complete lattice $L$ and a relational structure $\mathfrak{X}=(X,(R_i)_I)$, we introduce the convolution algebra $L^{\mathfrak{X}}$. This algebra consists of the lattice $L^X$ equipped with an additional $n_i$-ary operation $f_i$ for…
In this paper we derive structure theorems that characterize the spaces of linear and non-linear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful…
We study from a Hamiltonian point of view the generalized dispersionless KdV hierarchy of equations. From the so called dispersionless Lax representation of these equations we obtain three compatible Hamiltonian structures. The second and…
In a vertex algebra setting, we consider non-local screening operators associated to the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a…
We prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic $p$ which give sufficient conditions for the algebras to be of the form $[R^{(-)}, R^{(-)}] / (Z(R)…
A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…
We consider Fredholm determinants of matrix convolution operators associated to matrix versions of the $n - $th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlev\'e II hierarchy,…
A class of multidimensional integrable hierarchies connected with commutation of general (unreduced) (N+1)-dimensional vector fields containing derivative over spectral variable is considered. They are represented in the form of generating…
We develop efficient group-theoretical approach to the problem of classification of evolution equations that admit non-local transformation groups (quasi-local symmetries), i.e., groups involving integrals of the dependent variable. We…
Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for $(1+1)$-dimensional PDEs can be interpreted as ZCRs. In [arXiv:1303.3575], for any $(1+1)$-dimensional scalar…
We generate hierarchies of derivative nonlinear Schr\"odinger-type equations and their nonlocal extensions from Lie algebra splittings and automorphisms. This provides an algebraic explanation of some known reductions and newly established…
Tensor hierarchy algebras constitute a class of non-contragredient Lie superalgebras, whose finite-dimensional members are the "Cartan-type" Lie superalgebras in Kac's classification. They have applications in mathematical physics,…
We introduce a diagram category, study its structure, and investigate some of its applications to the representation theory of Lie algebras and Lie superalgebras. The morphisms of the category, which contains a subcategory isomorphic to the…
We study a quite general family of dynamical $r$-matrices for an auxiliary loop algebra ${\cal L}({su(2)})$ related to restricted flows for equations of the KdV type. This underlying $r$-matrix structure allows to reconstruct Lax…
In this work we develop a general procedure for constructing the recursion operators fro non-linear integrable equations admitting Lax representation. Svereal new examples are given. In particular we find the recursion operators for some…
Conformal algebras, recently introduced by Kac, encode an axiomatic description of the singular part of the operator product expansion in conformal field theory. The objective of this paper is to develop the theory of ``multi-dimensional''…
This paper addresses some fundamental issues in nonconvex analysis. By using pure complementary energy principle proposed by the author, a class of fully nonlinear partial diforerential equations in nonlinear elasticity is able to converted…
This paper establishes a categorical framework for phase spaces of Lie algebras, pre-Lie bialgebras, Manin triples, classical s-matrices, and relative Rota-Baxter operators by introducing the concept of coherent homomorphisms. Starting with…