Related papers: Classical Lie symmetries and reductions of a nonis…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
Singular perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact…
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…
We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of…
In this paper, we use various ansatzes with undetermined functions and the technique of moving frame to find solutions with parameter functions modulo the Lie point symmetries for the classical non-steady boundary layer problems. These…
This study uses Lie's theory of symmetries to compute the symmetry group of a class of partial differential equations parameterized by four constants: $u_{t}=-\left((a-bx)u_{x}+(d-ey)u_{y}+\frac{x}{2}u_{xx}+\frac{y}{2}u_{yy}\right)$; under…
An isomorphism between two hermitian unitals is proved, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).
A number of characteristics of integrable nonlinear partial differential equations (PDE's) for classical fields are reviewed, such as Backlund transformations, Lax pairs, and infinite sequences of conservation laws. An algebraic approach to…
Methods for the computation of invariants and symmetries of nonlinear evolution, wave, and lattice equations are presented. The algorithms are based on dimensional analysis, and can be implemented in any symbolic language, such as…
Many methods for reducing and simplifying differential equations are known. They provide various generalizations of the original symmetry approach of Sophus Lie. Plenty of relations between them have been noticed and in this note a unifying…
We perform a detailed classification of the Lie point symmetries and of the resulting similarity transformations for the Generalized Boiti-Leon-Pempinelli equations. The latter equations for a system of two nonlinear 1+2 partial…
We consider higher symmetries and operator symmetries of linear partial differential equations. The higher symmetries form a Lie algebra, and operator ones form an associative algebra. The relationship between these symmetries is…
We apply the Lie algebraic method to reflecting optical systems with plane-symmetric freeform mirrors. Using analytical ray-tracing equations we construct an optical map. The expansion of this map gives us the aberration coefficients in…
We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of…
In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic solutions. We perform the analysis using the Lie symmetry algebra…
We generalize the classical Lie results on a basis of differential invariants for a one-parameter group of local transformations to the case of arbitrary number of independent and dependent variables. It is proved that if universal…
Some connections between classical and nonclassical symmetries of a partial differential equation (PDE) are given in terms of determining equations of the two symmetries. These connections provide additional information for determining…
In the two papers of this series, we initiate the development of a new approach to implementing the concept of symmetry in classical field theory, based on replacing Lie groups/algebras by Lie groupoids/algebroids, which are the appropriate…
Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are…
Complete descriptions of the Lie symmetries of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity in one and two space dimensions are obtained. A surprisingly rich set of Lie symmetry algebras depending on…