Related papers: Integrable equations with negative evolution numbe…
Solvable structures are exploited in order to find families of explicit solutions to evolution PDEs admitting suitable differential constraints. The effectiveness of the method is verified on several explicit examples.
Obstacles to integrability in perturbed evolution equations are overcome by allowing higher-order terms in the expansion of the solution to depend explicitly on time and position. With a special expansion algorithm, obstacles vanish…
It is shown that a class of important integrable nonlinear evolution equations in (2+1) dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently, moving surfaces. Geometrical…
We consider an hierarchy of integrable 1+2-dimensional equations related to Lie algebra of the vector fields on the line. The solutions in quadratures are constructed depending on $n$ arbitrary functions of one argument. The most…
We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown…
Soliton equations in 2+1 and their 1+1 = 2+0 reductions are considered.
The paper is devoted to the study of annihilator extensions of evolution algebras and suggests an approach to classify finite-dimensional nilpotent evolution algebras. Subsequently nilpotent evolution algebras of dimension up to four are…
We consider here the class of fully-nonlinear symmetry-integrable third-order evolution equations in 1+1 dimensions that were proposed recently in the journal Open Communications in Nonlinear Mathematical Physics, vol. 2, 216--228 (2022).…
We show that the one dimensional unitary matrix model with potential of the form $a U + b U^2 + h.c.$ is integrable. By reduction to the dynamics of the eigenvalues, we establish the integrability of a system of particles in one space…
Exact solutions of Einstein's equations in 2+1-dimensional anti-de Sitter space containing any number of black holes are described. In addition to the black holes these spacetimes can possess ``internal'' structure. Accordingly the generic…
Integrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are…
This survey reviews variational and iterative methods for reconstructing non-negative solutions of ill-posed problems in infinite-dimensional spaces. We focus on two classes of methods: variational methods based on entropy-minimization or…
We provide a relation which describes how the entanglement of two d-level systems evolves as either system undergoes an arbitrary physical process. The dynamics of the entanglement turns out to be of a simple form, and is fully captured by…
We consider the cosmological models for the higher dimensional spacetime which includes the curvatures of our space as well as the curvatures of the internal space. We find that the condition for the integrability of the cosmological…
A systematic construction of a class of integrable hierarchy is discussed in terms of the twisted affine $A_{2r}^{(2)}$ Lie algebra. The zero curvature representation of the time evolution equations are shown to be classified according to…
Discusses several integrability tests for nonlinear evolution equations.
Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the…
We construct infinite dimensional families of non-singular stationary space times, solutions of the vacuum Einstein equations with a negative cosmological constant.
In this work we approach three-dimensional evolution algebras from certain constructions performed on two-dimensional algebras. More precisely, we provide four different constructions producing three-dimensional evolution algebras from…
We associate an square to any two dimensional evolution algebra. This geometric object is uniquely determined, does not depend on the basis and describes the structure and the behaviour of the algebra. We determine the identities of degrees…