Related papers: Nonlinear 2+1--Dimensional Field Equations from In…
We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related $R$-matrix theory for the Lie algebra of functions with respect…
The B\"acklund problem is solved for both the compact and noncompact versions of the Ishimori (2+1)-dimensional nonlinear spin model. In particular, a realization of the arising B\"acklund algebra in the form of an infinite-dimensional loop…
Three dimensional nonlinear wave interactions have been analytically described. The procedure under interest can be applied to three dimensional quasilinear systems of first order, whose hydrodynamic reductions are homogeneous…
We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that…
We extend a recently developed Hamiltonian formalism for nonlinear wave interaction processes in spatially periodic dielectric structures to the far-off-resonant regime, and investigate numerically the three-wave resonance conditions in a…
The analysis of nonlinear wave equations has experienced a dramatic growth in the last ten years or so. The key factor in this has been the transition from linear analysis, first to the study of bilinear and multilinear wave interactions,…
Nongraded infinite-dimensional Lie algebras appeared naturally in the theory of Hamiltonian operators, the theory of vertex algebras and their multi-variable analogues. They play important roles in mathematical physics. This survey article…
We apply the (direct and inverse) prolongation method to a couple of nonlinear Schr{\"o}dinger equations. These are taken as a laboratory field model for analyzing the existence of a connection between the integrability property and loop…
We consider some special type extensions of an arbitrary Lie algebra, which we call universal extensions. We show that these extensions are in one-to-one correspondence with finite dimensional associative commutative algebras. We also…
Symmetries for wave equation with additional conditions are found. Some conditions yield infinite-dimensional symmetry algebra for the nonlinear equation. Ansatzes and solutions corresponding to the new symmetries were constructed.
The cohomology and deformation theory of 3-Lie algebras are revisited. The theory of extending structures and unified product for 3-Lie algebras are developed.It is proved that the extending structures of 3-Lie algebras can be classified by…
We show that infinite families of non-relativistic spin-$3$ symmetries in $2+1$ dimensions, which include higher-spin extensions of the Bargmann, Newton-Hooke, non-relativistic Maxwell, and non-relativistic AdS-Lorentz algebras, can be…
Meta-conformal transformations are constructed as sets of time-space transformations which are not angle-preserving but contain time- and space translations, time-space dilatations with dynamical exponent ${z}=1$ and whose Lie algebras…
In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.
In an earlier work extensions of supersymmetry and super Lie algebras were constructed consistently starting from any representation $\D$ of any Lie algebra $\g$. Here it is shown how infinite dimensional Lie algebras appear naturally…
We provide a Lie algebra expansion procedure to construct three-dimensional higher-order Schr\"odinger algebras which relies on a particular subalgebra of the four-dimensional relativistic conformal algebra. In particular, we reproduce the…
A constructive procedure is proposed for formulation of linear differential equations invariant under global symmetry transformations forming a semi-simple Lie algebra f. Under certain conditions f-invariant systems of differential…
Nonlinear Dirac equations in D+1 space-time are obtained by variation of the spinor action whose Lagrangian components have the same conformal degree and the coupling parameter of the self-interaction term is dimensionless. In 1+1…
We introduce the concept of a triangular representation of a Lie algebra, give a counterpart of Ado's theorem, and discuss $2$-irreducible triangular modules over a nonreductive Lie algebra.
The relation between nonlinear algebras and linear ones is established. For one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to…