Method Of Group Foliation Research Articles

Group foliation of finite difference equations

Using the theory of equivariant moving frames, a group foliation method for invariant finite difference equations is developed. This method is analogous to the group foliation of differential equations and uses the symmetry group of the equation to decompose the solution process into two steps, called resolving and reconstruction. Our constructions are performed algorithmically and symbolically by making use of discrete recurrence relations among joint invariants. Applications to invariant finite difference equations that approximate differential equations are given.

Conformal invariance and new exact solutions of the elastostatics equations

We fulfilled a group foliation of the system of n-dimensional (n ≥ 2) Lame equations of the classical static theory of elasticity with respect to the infinite subgroup contained in normal subgroup of main group of this system. It permitted us to move from the Lame equations to the equivalent unification of two first-order systems: automorphic and resolving. We obtained a general solution of the automorphic system. This solution is an n-dimensional analogue of the Kolosov-Muskhelishvili formula. We found the main Lie group of transformations of the resolving system of this group foliation. It turned out that in the two-dimensional and three-dimensional cases, which have a physical meaning, this system is conformally invariant, while the Lame equations admit only a group of similarities of the Euclidean space. This is a big success, since in the method of group foliation, resolving equations usually inherit Lie symmetries subgroup of the full symmetry group that was not used for the foliation. In the three-dimensional case for the solutions of the resolving system, we found the general form of the transformations similar to the Kelvin transformation. These transformations are the consequence of the conformal invariance of the resolving system. In the three-dimensional case with a help of the complex dependent and independent variables, the resolving system is written as a simple complex system. This allowed us to find non-trivial exact solutions of the Lame equations, which direct for the Lame equations practically impossible to obtain. For this complex system, all the essentially distinct invariant solutions of the maximal rank we have found in explicit form, or we reduced the finding of those solutions to the solving of the classical one-dimensional equations of the mathematical physics: the heat equation, the telegraph equation, the Tricomi equation, the generalized Darboux equation, and other equations. For the resolving system, we obtained double wave of a special type and double waves of the shear deformations in an elastic medium. By the obtained formulas for generating new solutions, these solutions give an infinite set of exact solutions of this complex system. By the three-dimensional analog of the Kolosov-Muskhelishvili formula, the found solutions produce an infinite-dimensional vector space of exact solutions of the Lame equations.

GROUP FOLIATION OF DIFFERENTIAL EQUATIONS USING MOVING FRAMES

We incorporate the new theory of equivariant moving frames for Lie pseudogroups into Vessiot’s method of group foliation of differential equations. The automorphic system is replaced by a set of reconstruction equations on the pseudogroup jets. The result is a completely algorithmic and symbolic procedure for finding both invariant and noninvariant solutions of differential equations admitting a symmetry group.

A family of heavenly metrics

This is a corrected and essentially extended version of the unpublished manuscript by Y Nutku and M Sheftel which contains new results. It is proposed to be published in honour of Y Nutku’s memory. All corrections and new results in sections 1, 2 and 4 are due to M Sheftel. We present new anti-self-dual exact solutions of the Einstein field equations with Euclidean and neutral (ultra-hyperbolic) signatures that admit only one rotational Killing vector. Such solutions of the Einstein field equations are determined by non-invariant solutions of Boyer–Finley (BF) equation. For the case of Euclidean signature such a solution of the BF equation was first constructed by Calderbank and Tod. Two years later, Martina, Sheftel and Winternitz applied the method of group foliation to the BF equation and reproduced the Calderbank–Tod solution together with new solutions for the neutral signature. In the case of Euclidean signature we obtain new metrics which asymptotically locally look like a flat space and have a non-removable singular point at the origin. In the case of ultra-hyperbolic signature there exist three inequivalent forms of metric. Only one of these can be obtained by analytic continuation from the Calderbank–Tod solution whereas the other two are new.

Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors

We demonstrate how a combination of our recently developed methods of partner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge-Amp\`ere equation (CMA) and provide a lift from invariant solutions of CMA satisfying Boyer-Finley equation to non-invariant ones. Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat anti-self-dual Einstein-K\"ahler metric with Euclidean signature without Killing vectors, together with Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very special choices of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in a bounded domain.

Group foliation of equations in geophysical fluid dynamics

The method of group foliation can be used to construct solutions to a system of partial differential equations that, as opposed to Lie's method of symmetry reduction, are not invariant under any symmetry of the equations. The classical approach is based on foliating the space of solutions into orbits of the given symmetry group action, resulting in rewriting the equations as a pair of systems, the so-called automorphic and resolvent systems, involving the differential invariants of the symmetry group, while a more modern approach utilizes a reduction process for an exterior differential system associated with the equations. In each method solutions to the reduced equations are then used to reconstruct solutions to the original equations. We present an application of the two techniques to the one-dimensional Korteweg-de Vries equation and the two-dimensional Flierl-Petviashvili (FP) equation. An exact analytical solution is found for the radial FP equation, although it does not appear to be of direct geophysical interest.

Functional Separable Solutions of Nonlinear Heat Equations in Non-Newtonian Fluids

We study the functional separation of variables to the nonlinear heat equation: ut = (A(x)D(u)uxn)x + B(x)Q(u), Ax ≠ 0. Such equation arises from non-Newtonian fluids. Its functional separation of variables is studied by using the group foliation method. A classification of the equation which admits the functional separable solutions is performed. As a consequence, some solutions to the resulting equations are obtained.

Lift of noninvariant solutions of heavenly equations from three to four dimensions and new ultra-hyperbolic metrics

We demonstrate that partner symmetries provide a lift of noninvariant solutions of the three-dimensional Boyer–Finley equation to noninvariant solutions of the four-dimensional hyperbolic complex Monge–Ampère equation. The lift is applied to noninvariant solutions of the Boyer–Finley equation, obtained earlier by the method of group foliation, to yield noninvariant solutions of the hyperbolic complex Monge–Ampère equation. Using these solutions we construct new Ricci-flat ultra-hyperbolic metrics with non-zero curvature tensor that have no Killing vectors.

Functional Separable Solutions to NonlinearDiffusion Equations by Group Foliation Method

We consider the functional separation of variables to the nonlinear diffusionequation with source and convection term:ut = (A(x)D(u)ux)x+B(x)Q(u), Ax≠0. The functionalseparation of variables to this equation is studied by using thegroup foliation method. A classification is carried out for theequations which admit the function separable solutions. As aconsequence, some solutions to the resulting equations are obtained.

Functionally separable solutions to nonlinear wave equations by group foliation method

We apply functional separation of variables within the approach of the group foliation method to the nonlinear wave equation with variable speed and external force: u t t = A ( x ) ( D ( u ) u x ) x + B ( x ) Q ( u ) , A x ≠ 0 . A classification of these equations admitting functionally separable solutions is performed and the resulting solutions are obtained in explicit form in many cases.

Extended Group Foliation Method and Functional Separationof Variables to Nonlinear Wave Equations**The projectsupported by National Natural Science Foundation of Chinaunder Grant No. 10371098, the Natural Science Foundationof Shanxi Province of China, and the Excellent Young Teachers' Foundationof the Ministry of Education of China

Generalized functional separation of variables tononlinear evolution equations is studied in terms of the extendedgroup foliation method, which is based on the Lie point symmetrymethod. The approach is applied to nonlinear wave equations withvariable speed and external force. A complete classification forthe wave equation which admits functional separable solutions ispresented. Some known results can be recovered by this approach.

Group Foliation Method and Functional Separation of Variablesto Nonlinear Diffusion Equations

Generalized functional separation of variables to nonlinear diffusionequations is studied in terms of the extended group foliation method. Acomplete classification for the nonlinear diffusion equation with sourceterm which admits functional separable solutions is presented.

Method of Group Foliation, Hodograph Transformation, and Noninvariant Solutions of the Heavenly Equation

We present the method of group foliation for constructing noninvariant solutions of partial differential equations on an important example of the “heavenly equation” from the theory of gravitational instantons. We show that the constraint of commutativity of a pair of invariant differential operators leads to a set of noninvariant solutions of the heavenly equation. In the second part of the paper, we demonstrate how the noninvariant solution of the ultrahyperbolic heavenly equation recently obtained by Mañas and Martínez Alonso becomes obvious after hodograph transformation of the heavenly equation. Because of extra symmetries, this solution is conditionally invariant, unlike noninvariant solutions obtained previously. We make the hodograph transformation of the group foliation structure and derive two invariant relations valid for the hodograph solution, in addition to resolving equations in an attempt to obtain the orbit of this solution.

Method of group foliation and non-invariant solutions of partial differential equations

Using the heavenly equation as an example, we propose the method of group foliation as a tool for obtaining non-invariant solutions of PDEs with infinite-dimensional symmetry groups. The method involves the study of compatibility of the given equations with a differential constraint, which is automorphic under a specific symmetry subgroup and therefore selects exactly one orbit of solutions. By studying the integrability conditions of this automorphic system, i.e. the resolving equations, one can provide an explicit foliation of the entire solution manifold into separate orbits. The new important feature of the method is the extensive use of the operators of invariant differentiation for the derivation of the resolving equations and for obtaining their particular solutions. Applying this method we obtain exact analytical solutions of the heavenly equation, non-invariant under any subgroup of the symmetry group of the equation.

Application of the group foliation method to the complex Monge-Ampère equation

We apply the method of group foliation to the complex Monge-Ampere equation (CMA2) to establish a regular framework for finding its non-invariant solutions. We employ an infinite symmetry subgroup ofCMA2 to produce a foliation of the solution space into orbits of solutions with respect to this group and a corresponding splitting ofCMA2 into an automorphic system and a resolvent system. We propose a new approach to group foliation which is based on the commutator algebra of operators of invariant differentiation. This algebra together with its Jacobi identities provides the commutator representation of the resolvent system.

Group Foliation Approach to the Complex Monge–Ampère Equation

We apply the group foliation method to find noninvariant solutions of the complex Monge–Ampere equation (CMA2). We use the infinite symmetry subgroup of the CMA2 to foliate the solution space into orbits of solutions with respect to this group and correspondingly split the CMA2 into an automorphic system and a resolvent system. We propose a new approach to group foliation based on the commutator algebra of operators of invariant differentiation. This algebra together with Jacobi identities provides the commutator representation of the resolvent system. For solving the resolvent system, we propose symmetry reduction, which allows deriving reduced resolving equations.

Differential invariants and group foliation for the complex Monge-Ampère equation

We apply the method of group foliation to the complex Monge-Ampere equation ({CMA}2) with the goal of establishing a regular framework for finding its non-invariant solutions. We employ the infinite symmetry subgroup of the equation, the group of unimodular biholomorphisms, to produce a foliation of the solution space into leaves which are orbits of solutions with respect to the symmetry group. Accordingly, {CMA}2 is split into an automorphic system and a resolvent system which we derive in this paper. This is an intricate system and here we make no attempt to solve it in order to obtain non-invariant solutions. We obtain all differential invariants up to third order for the group of unimodular biholomorphisms and, in particular, all the basis differential invariants. We construct the operators of invariant differentiation from which all higher differential invariants can be obtained. Consequently, we are able to write down all independent partial differential equations with one real unknown and two complex independent variables which keep the same infinite symmetry subgroup as {CMA}2. We prove explicitly that applying operators of invariant differentiation to third-order invariants we obtain all fourth-order invariants. At this level we have all the information which is necessary and sufficient for group foliation. We propose a new approach in the method of group foliation which is based on the commutator algebra of operators of invariant differentiation. The resolving equations are obtained by applying this algebra to differential invariants with the status of independent variables. Furthermore, this algebra together with Jacobi identities provides the commutator representation of the resolvent system. This proves to be the simplest and most natural way of arriving at the resolving equations.