Lie Group Action. A process by which a Lie group, acting as a symmetry, moves points in a space. When points in the space that are related by a group element are identified, one obtains the quotient space. Free Action. An action that moves every point under any nontrivial group element. Proper Action. An action that obeys a compactness condition. Momentum Mapping. A dynamically conserved quanti...

Lie symmetry analysis is an established method for generating symmetries of differential equations. We apply this together the generalized fundamental theorem double reduction. In particular, Noether and some associated conservation laws are constructed in our investigation to find exact solutions higher order partial equations complex

In this paper we study symmetry reductions of a class of nonlinear fourth order partial differential equations utt = ( κu+ γu ) xx + uuxxxx + μuxxtt + αuxuxxx + βu 2 xx , (1) where α, β, γ, κ and μ are arbitrary constants. This equation may be thought of as a fourth order analogue of a generalization of the Camassa-Holm equation, about which there has been considerable recent interest. Further ...

Abstract: A conservation law theorem stated by N. Ibragimov along with its subsequent extensions are shown to be a special case of a standard formula that uses a pair consisting of a symmetry and an adjoint-symmetry to produce a conservation law through a well-known Fréchet derivative identity. Furthermore, the connection of this formula (and of Ibragimov’s theorem) to the standard action of sy...

We introduce an extension to local principal component analysis for learning symmetric manifolds. In particular, we use a spectral method approximate the Lie algebra corresponding symmetry group of underlying manifold. derive sample complexity our various manifolds before applying it data sets improved density estimation.

We establish an algebraic structure for zero curvature representations of coupled integrable couplings. The adopted zero curvature representations are associated with Lie algebras possessing two sub-Lie algebras in form of semi-direct sums of Lie algebras. By applying the presented algebraic structures to the AKNS systems, we give an approach for generating τ -symmetry algebras of coupled integ...

Using Lie symmetry methods for differential equations we have investigated the symmetries of a Lagrangian for a plane symmetric static spacetime. Perturbing this Lagrangian we explore its approximate symmetries. It has a non-trivial first-order approximate symmetry.

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We i...

We show that any Lie point symmetry of semilinear Kohn-Laplace equations on the Heisenberg group H with power nonlinearity is a divergence symmetry if and only if the corresponding exponent assumes critical value.