We extend our analysis on the Lie symmetries in fluid dynamics to case of macroscopic traffic estimation models. In particular we study Aw–Rascle–Zhang model for estimation, which consists two hyperbolic first-order partial differential equations. The symmetries, one-dimensional optimal system and corresponding invariants are determined. Specifically, find that admitted form four-dimensional al...

Differential-difference equations (DDEs) of the form u n (t) = Fn(t, un+a, . . . , un+b) with k ≥ 2 are studied for Lie symmetries and preliminary classification. Explicit forms of equations are given for those admitting at least one intrinsic Lie symmetry. An algorithmic mechanism is also proposed to automate the symmetry calculation for fairly general DDEs via computer algebras.

We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.

In a recent paper Carot et al. considered the definition of cylindrical symmetry as a specialisation of the case of axial symmetry. One of their propositions states that if there is a second Killing vector, which together with the one generating the axial symmetry, forms the basis of a two-dimensional Lie algebra, then the two Killing vectors must commute, thus generating an Abelian group. In t...

In the paper, methods of Lie group analysis are applied to investigate symmetry properties of some classes of nonlinear fractional differential equations. For class of equations Dy = f(x, y), 0 < α < 1, the problem of group classification is solved. Symmetry properties of equations D t u = (k(u)ux)x, 0 < α ≤ 2 for different orders of α are compared. Obtained symmetries are used to construct exa...

Abstract: A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is prese...

In this article we study a new kind of unbounded solutions to the Novikov equation, found via Lie symmetry analysis. These exhibit peakon creation, i.e., these are smooth up u ...

The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical Lie–Bäcklund generator is extended to include any Lie–Bäcklund generator. Also, it is shown that the Lie algebra of Lie–Bäcklund symmetries of a conserved vector of a system is a subalgebra of the Lie–Bäcklund symmetries of the system. Moreover, we in...

Spinor representation of the group GL(4,R) is needed for correct description of the Fermi fields on Riemann space, such as the space-time of general relativity. It is used for two purposes: to define the connectivity and covariant derivative of spinor field and to define the Lie derivative. Recent publications [1–3] have reminded of this problem. The important problem for definition of Fermi fi...

We construct a new class of exact solutions describing spacetimes possessing Lie algebroid symmetry. They are described by generic off– diagaonal 5D metrics embedded in bosonic string gravity and possess nontrivial limits to the Einstein gravity. While we focus on nonholonomic vielbein transforms of the Schwarzschild metrics to 5D ansatz with solitonic backgrounds, much of the analysis continue...