Continuous and discrete symmetries of renormalization group equations for neutrino oscillations in matter

Three-flavor neutrino oscillations in matter can be described by three effective neutrino masses mi (for i = 1, 2, 3) and the effective mixing matrix Vαi (for α = e, µ, τ and i = 1, 2, 3). When the matter parameter a ≡ 2√2GFNeE is taken as an independent variable, a complete set of first-order ordinary differential equations for m2 i and |Vαi|2have been derived in the previous works. In the present paper, we point out that such a system of differential equations possesses both the continuous symmetries characterized by one-parameter Lie groups and the discrete symmetry associated with the permutations of three neutrino mass eigenstates. The implications of these symmetries for solving the differential equations and looking for differential invariants are discussed.

Lie Symmetry Analysis, Exact Solutions, and Conservation Laws of Variable-Coefficients Boiti-Leon-Pempinelli Equation

In this article, we study the generalized ( 2 + 1 )-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the ( 1 + 1 )-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through G ′ / G -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method.

Classification of singular differential invariants in (1+3)-dimensional space and integrability

Singularity is one of the important features in invariant structures in several physical phenomena reflected often in the associated invariant differential equations. The classification problem for singular differential invariants in (1+3)-dimensional space associated with Lie algebras of dimension 4 is investigated. The formulation of singular invariants for a Lie algebra of dimension [Formula: see text] possessed by the underlying system of three second-order ordinary differential equations is studied in detail and the corresponding canonical forms for these systems are deduced. Furthermore, the categorization of singular invariants on the basis of conditional singularity, weak uncoupling, weak linearization, partial uncoupling and partial linearization are described for the underlying canonical forms. In addition, those cases of classified canonical forms are also mentioned which do not lead to singular invariant systems of three second-order ODEs for a Lie algebra of dimension 4. The integrability aspect of these classified singular-invariant systems in (1+3)-dimensional space is discussed in a detailed manner for a Lie algebra of dimension 4. Finally, two physical systems from mechanics are presented to illustrate the utilization of the physical aspect of these singular invariants.

On Metric Invariants of Spherical Harmonics

The algebraic and differential SO3-invariants of spherical harmonics are studied in this work. The fields of rational algebraic and rational differential invariants and their applications for the description of regular SO3-orbits of spherical harmonics are given.

Invariants of Surfaces in Three-Dimensional Affine Geometry

Using the method of moving frames we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is non-trivial, then it is generically generated by a single invariant.

Riemann problem for rate-type materials with non-constant initial conditions

In this paper, using the compatible theory of differential invariants, a class of exact solutions is obtained for nonhomogeneous quasilinear hyperbolic system of partial differential equations (PDEs) describing rate type materials; these solutions exhibit genuine nonlinearity that leads to the formation of discontinuities such as shocks and rarefaction waves. For certain nonconstant and smooth initial data, the solution to the Riemann problem is presented providing a complete characterisation of the solutions.

Lagrangian Curve Flows on Symplectic Spaces

A smooth map γ in the symplectic space R2n is Lagrangian if γ,γx,…, γx(2n−1) are linearly independent and the span of γ,γx,…,γx(n−1) is a Lagrangian subspace of R2n. In this paper, we (i) construct a complete set of differential invariants for Lagrangian curves in R2n with respect to the symplectic group Sp(2n), (ii) construct two hierarchies of commuting Hamiltonian Lagrangian curve flows of C-type and A-type, (iii) show that the differential invariants of solutions of Lagrangian curve flows of C-type and A-type are solutions of the Drinfeld-Sokolov’s C^n(1)-KdV flows and A^2n−1(2)-KdV flows respectively, (iv) construct Darboux transforms, Permutability formulas, and scaling transforms, and give an algorithm to construct explicit soliton solutions, (v) give bi-Hamiltonian structures and commuting conservation laws for these curve flows.