Sophus Lie developed a systematic way to solve ODEs. He found that transformations which form a continuous group and leave a differential equation invariant can be used to simplify the equation. Lie’s method uses the infinitesimal generator of these point transformations. These are symmetries of the equation mapping solutions into solutions. Lie’s methods did not find widespread use in part bec...

We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation, which has the classical Boussinesq equation as an special case. We study the class of functions f(u) for which this equation admit either the classical or the nonclassical method. The reductions obtained are derived. Some new exact solutions can be der...

We prove that the extended Toda hierarchy of [1] admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators Lm, m ≥ −1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination...

The N-extended self-dual supergravity in the ultra-hyperbolic four-dimensional space-time of kleinian signature (2+2) is given in the N-extended harmonic superspace. We reformulate the on-shell N-extended self-dual supergravity constraints of Siegel to a 'zero-curvature' representation, and solve all of them but one in terms of a single superfield prepotential, by using a covariant Frobenius ga...

The level crossing problem is neatly formulated by the second quantized formulation, which exhibits a hidden local gauge symmetry. The analysis of geometric phases is reduced to a simple diagonalization of the Hamiltonian. If one diagonalizes the geometric terms in the infinitesimal neighborhood of level crossing, the geometric phases become trivial (and thus no monopole singularity) for arbitr...

We prove that particle models whose action is given by the integrated n-th curvature function over the world line possess n+ 1 gauge invariances. A geometrical characterization of these symmetries is obtained via Frenet equations by rephrasing the n-th curvature model in Rd in terms of a standard relativistic particle in Sd−n. We “prove by example” that the algebra of these infinitesimal gauge ...

We discuss the relation between space–time diffeomorphisms and gauge transformations in theories of the Yang–Mills type coupled with Einstein’s general relativity. We show that local symmetries of the Hamiltonian and Lagrangian formalisms of these generally covariant gauge systems are equivalent when gauge transformations are required to induce transformations which are projectable under the Le...

The vacuum overlap formalism is extended to describe the supersymmetric multiplet of a Weyl fermion, a complex scalar boson and an auxiliary field in the case without interaction, based on the fact that supersymmetry can be maintained upto quadratic terms by introducing bosonic species doublers. We also obtain a local action which describes the chiral multiplet and discuss its symmetry structur...

We show that certain infinitesimal operators of the Lie–point symmetries of the incompressible 3D Navier–Stokes equations give rise to vortex solutions with different characteristics. This approach allows an algebraic classification of vortices and throws light on the alignment mechanism between the vorticity → ω and the vortex stretching vector S → ω, where S is the strain matrix. The symmetry...

Noncommutative geometry is an established potential candidate for including quantum phenomena in gravitation. We outline the formalism of Hopf algebras and its connection to algebra infinitesimal diffeomorphisms. Using a Drinfeld twist we deform spacetime symmetries, vector fields differential forms leading formulation noncommutative Einstein equations. study concrete example charged BTZ deform...