Symmetry Pseudo-groups Research Articles

Extended symmetry analysis of (1+2)‐dimensional fine Kolmogorov backward equation

AbstractWithin the class of (1+2)‐dimensional ultraparabolic linear equations, we distinguish a fine Kolmogorov backward equation with a quadratic diffusivity. Modulo the point equivalence, it is a unique equation within the class whose essential Lie invariance algebra is five‐dimensional and nonsolvable. Using the direct method, we compute the point symmetry pseudogroup of this equation and analyze its structure. In particular, we single out its essential subgroup and classify its discrete elements. We exhaustively classify all subalgebras of the corresponding essential Lie invariance algebra up to inner automorphisms and up to the action of the essential point‐symmetry group. This allowed us to classify Lie reductions and Lie invariant solutions of the equation under consideration. We also discuss the generation of its solutions using point and linear generalized symmetries and carry out its peculiar generalized reductions. As a result, we construct wide families of its solutions parameterized by an arbitrary finite number of arbitrary solutions of the (1+1)‐dimensional linear heat equation or one or two arbitrary solutions of (1+1)‐dimensional linear heat equations with inverse square potentials.

Extended symmetry analysis of remarkable (1+2)-dimensional Fokker–Planck equation

AbstractWe carry out the extended symmetry analysis of an ultraparabolic Fokker–Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker–Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker–Planck equation using the direct method, analyse its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalised reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker–Planck equation, in particular, those parameterised by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker–Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.

On Lie’s problem and differential invariants of ODEs y″ = F(x, y)

A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.

Deformed cohomologies of symmetry pseudo-groups and coverings of differential equations

Abstract The work establishes a relation between deformed cohomologies of symmetry pseudo-groups and coverings of differential equations. Examples include the potential Khokhlov–Zabolotskaya equation and the Boyer–Finley equation.

Differential Invariants of the (2+1)-Dimensional Breaking Soliton Equation

Abstract We construct the differential invariants of Lie symmetry pseudogroups of the (2+1)-dimensional breaking soliton equation and analyze the structure of the induced differential invariant algebra. Their syzygies and recurrence relations are classified. In addition, a moving frame and the invariantization of the breaking soliton equation are also presented. The algorithms are based on the method of equivariant moving frames.

Integrable Dispersionless PDEs in 4D, Their Symmetry Pseudogroups and Deformations

We study integrable non-degenerate Monge-Ampere equations of Hirota type in 4D and demonstrate that their symmetry algebras have a distinguished graded structure, uniquely determining the equations. This is used to deform these heavenly type equations into new integrable PDE of the second order with large symmetry pseudogroups. We classify the obtained symmetric deformations and discuss self-dual hyper-Hermitian geometry of their solutions, which encode integrability via the twistor theory.

Symmetries of almost complex structures and pseudoholomorphic foliations

Contrary to complex structures, a generic almost complex structure J has no local symmetries. We give a criterion for finite-dimensionality of the pseudogroup G of local symmetries for a given almost complex structure J. It will be indicated that a large symmetry pseudogroup (infinite-dimensional) is a signature of some integrable structure, like a pseudoholomorphic foliation. We classify the sub-maximal (from the viewpoint of the size of G) symmetric structures J. Almost complex structures in dimensions 4 and 6 are discussed in greater details in this paper.

The four-dimensional Martínez Alonso–Shabat equation: Differential coverings and recursion operators

We apply Cartan's method of equivalence to find a contact integrable extension for the structure equations of the symmetry pseudo-group of the four-dimensional Martinez Alonso - Shabat equation. From the extension we derive two differential coverings including coverings with one and two non-removable parameters. Then we apply the same approach to construct a recursion operator for symmetries of the equation under study.

Contact integrable extensions of symmetry pseudo-groups and coverings of (2+1 ) dispersionless integrable equations

We apply the technique of integrable extensions to the symmetry pseudo-groups of the r -th mdKP equation, the r -th dDym equation, and the deformed Boyer–Finley equation. This gives another look at deriving known coverings and allows us to find new coverings for these equations.

Cartan’s Structure of Symmetry Pseudo-Group and Coverings for the r-th Modified Dispersionless Kadomtsev–Petviashvili Equation

We derive two non-equivalent coverings for the r-th dKP equation from Maurer–Cartan forms of its symmetry pseudo-group. Also we find Bäcklund transformations between the obtained covering equations.

New classification techniques for ordinary differential equations

The goal of the present paper is to propose an enhanced ordinary differential equation solver by exploitation of the powerful equivalence method of Élie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of a dictionary of ODEs, that are regarded as well-known, or at least well-studied. The dictionary considered in this article comprises the ODEs in a book of Kamke. The major advantage of our solver is that the equivalence transformation is obtained without integrating differential equations. We provide also a theoretical contribution revealing the relationship between the change of coordinates that maps two differential equations and their symmetry pseudo-groups.

Élie Cartan’s geometrical vision or how to avoid expression swell

The aim of the paper is to demonstrate the superiority of Cartan’s method over direct methods based on differential elimination for handling otherwise intractable equivalence problems. In this sense, using our implementation of Cartan’s method, we establish two new equivalence results. We establish when a system of second order ODEs is equivalent to flat system (second derivations are zero), and when a system of holomorphic PDEs with two independent variables and one dependent variable is flat. We consider the problem of finding transformation that brings a given equation to the target one. We shall see that this problem becomes algebraic when the symmetry pseudogroup of the target equation is zerodimensional. We avoid the swelling of the expressions, by using non-commutative derivations adapted to the problem.

Cartan’s Structure Theory of Symmetry Pseudo-Groups, Coverings and Multi-Valued Solutions for the Khokhlov–Zabolotskaya Equation

We derive two non-equivalent coverings for the modified Khokhlov–Zabolotskaya equation from Maurer–Cartan forms of its symmetry pseudo-group. Also we find Backlund transformations between the obtained covering equations. We apply these results to constructing multi-valued solutions for the Khokhlov–Zabolotskaya equation.

Coverings of Differential Equations and Cartan’s Structure Theory of Lie Pseudo-Groups

We establish relations between Maurer–Cartan forms of symmetry pseudo-groups and coverings of differential equations. Examples include Liouville’s equation, the Khokhlov–Zabolotskaya equation, and the Boyer–Finley equation.

Algorithms for Differential Invariants of Symmetry Groups of Differential Equations

We develop new computational algorithms, based on the method of equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudo-groups of differential equations and analyzing the structure of the induced differential invariant algebra. The Korteweg-deVries (KdV) and Kadomtsev-Petviashvili (KP) equations serve to illustrate examples. In particular, we deduce the first complete classification of the differential invariants and their syzygies of the KP symmetry pseudo-group.

Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations

A new method of constructing structure equations of Lie symmetry pseudogroups of differential equations, dispensing with explicit solutions of the (infinitesimal) determining systems of the pseudogroups, is presented, and illustrated by the examples of the Kadomtsev–Petviashvili and Korteweg–de Vries equations.

Moving coframes and symmetries of differential equations

We formulate a method of computing invariant 1-forms and structure equations of symmetry pseudo-groups of differential equations based on Cartan's method of equivalence and the moving coframe method introduced by Fels and Olver. Our apparoach does not require a preliminary computation of infinitesimal defining systems, their analysis and integration, and uses differentiation and linear algebra operations only. Examples of its applications are given.

Geometry and Structure of Lie Pseudogroups from Infinitesimal Defining Systems

An algorithm is described which uses a finite number of differentiations and linear operations to determine the Cartan structure of a transitive Lie pseudogroup from its infinitesimal defining equations. In addition, an algorithm is presented for determining from the infinitesimal defining system whether a Lie pseudogroup has essential invariants. If such invariants exist, the pseudogroup is intransitive. These methods make feasible the calculation of the Cartan structure of infinite Lie pseudogroups of symmetries of differential equations. The structure of the symmetry pseudogroup of the KP equation is presented.