相关论文: Group Invariant Solutions in Mathematical Physics …

We present a generalization of Lie's method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation…

Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…

A full Lie point symmetry analysis of rational difference equations is performed. Non-trivial symmetries are derived and exact solutions using these symmetries are obtained.

We extend Lie's classical method for finding group invariant solutions to the case of non-transverse group actions. For this extension of Lie's method we identify a local obstruction to the principle of symmetric criticality. Two examples…

In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form…

The description of invariants of surfaces with respect to the motion groups is reduced to the description of invariants of parameterized surfaces with respect to the motion groups. Existence of a commuting system of invariant partial…

We study invariance properties of Colombeau generalized functions under actions of smooth Lie transformation groups. Several characterization results analogous to the smooth setting are derived and applications to generalized rotational…

We prove an equivariant implicit function theorem for variational problems that are invariant under a varying symmetry group (corresponding to a bundle of Lie groups). Motivated by applications to families of geometric variational problems…

The use of geometric invariants has recently played an important role in the solution of classification problems in non-commutative ring theory. We construct geometric invariants of non-commutative projectivizations, a significant class of…

Physical theories grounded in mathematical symmetries are an essential component of our understanding of a wide range of properties of the universe. Similarly, in the domain of machine learning, an awareness of symmetries such as rotation…

We study generalized group actions on differentiable manifolds in the Colombeau framework, extending previous work on flows of generalized vector fields and symmetry group analysis of generalized solutions. As an application, we analyze…

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the…

The concept of the theory of continuous groups of transformations has attracted the attention of applied mathematicians and engineers to solve many physical problems in the engineering sciences. Three applications are presented in this…

We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the…

This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130,…

We present a new approach to the proof of ergodic theorems for actions of free groups based on geometric covering and asymptotic invariance arguments. Our approach can be viewed as a direct generalization of the classical geometric covering…

74J30The maximal group of Lie point symmetries of a system of nonlinear equations used in geophysical fluid dynamics is presented. The Lie algebra of this group is infinite-dimensional and involves three arbitrary functions of time. The…

Differential invariants of a (pseudo)group action can vary when restricted to invariant submanifolds (differential equations). The algebra is still governed by the Lie-Tresse theorem, but may change a lot. We describe in details the case of…

Using the theory of the symmetry group for PDEs [15, 17], we derive the symmetry group G associated to surfaces PDE. Several group invariant solutions of the surfaces PDE are given by solving a reduced system of partial differential…

The main physical result of this paper are exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation.…