Related papers: Lie symmetries for systems of evolution equations
The concept of symmetries in physics is briefly reviewed. In the first part of these lecture notes, some of the basic mathematical tools needed for the understanding of symmetries in nature are presented, namely group theory, Lie groups and…
We exhaustively classify the Lie reductions of the real dispersionless Nizhnik equation to partial differential equations in two independent variables and to ordinary differential equations. Lie and point symmetries of reduced equations are…
Symmetry is a powerful tool for finding analytical solutions to differential equations, both partial and ordinary, via the similarity variables or via the invariance of the equation under group transformations. It is the largest group of…
On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including…
An alternative approach - nonequilibrium evolution thermodynamics, is compared with classical Landau approach. A statistical justification of the approach is carried out with help of probability distribution function on an example of a…
We discuss various compatibility criteria for overdetermined systems of PDEs generalizing the approach to formal integrability via brackets of differential operators. Then we give sufficient conditions that guarantee that a PDE possessing a…
Lie symmetry group method is applied to study the boundary-layer equations for two-dimensional steady flow of an incompressible, viscous fluid near a stagnation point at a heated stretching sheet placed in a porous medium equation. The…
We consider four dimensional Lie groups with left-invariant Riemannian metrics. For such groups we classify left-invariant conformal foliations with minimal leaves of codimension two. These foliations produce local complex-valued harmonic…
The Liouville equation is well known to be linearizable by a point transformation. It has an infinite dimensional Lie point symmetry algebra isomorphic to a direct sum of two Virasoro algebras. We show that it is not possible to discretize…
We classify, up to automorphism, left invariant Riemannian metrics on 4-dimensional simply connected nonunimodular Lie groups. This is equivalent to classifying, up to automorphism, inner products on 4-dimensional nonunimodular Lie…
In this paper we use the $Z-$decomposition as a tool to find locally symmetric left invariant Riemannian metrics on some Lie groups. For this purpose, we need to compute the spectrum of the curvature operator. Since the study of this…
The strict connection between Lie point-symmetries of a dynamical system and its constants of motion is discussed and emphasized, through old and new results. It is shown in particular how the knowledge of a symmetry of a dynamical system…
We study Lie point symmetry structure of generalized nonlinear wave equations in the $(n+1)$-dimensional space-time.
It is pointed out that affine Lie algebras appear to be the natural mathematical structure underlying the notion of integrability for two-dimensional systems. Their role in the construction and classification of 2D integrable systems is…
We consider stochastic equations for the class of formal mappings. Existence and uniqueness of solution, as well as evolution property are proved.
We determine the Lie point symmetries of the Schr\"{o}dinger and the Klein Gordon equations in a general Riemannian space. It is shown that these symmetries are related with the homothetic and the conformal algebra of the metric of the…
In this paper, we address several interconnected problems in the theory of harmonic maps between Riemannian manifolds. First, we present necessary background and establish one of the main results of the paper: a criterion characterizing…
The first part of the book is devoted to the symmetry approach to classification of scalar integrable evolution PDEs with two independent variables. In the second part systems of evolution equations with polynomial homogeneous right-hand…
We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the…
Multisymplectic variational integrators are structure preserving numerical schemes especially designed for PDEs derived from covariant spacetime Hamilton principles. The goal of this paper is to study the properties of the temporal and…