Francesco Oliveri

Lie Symmetries of Differential Equations: Classical Results and Recent Contributions

Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems, to the derivation of conservation laws, to the construction of links between different differential equations that turn out to be equivalent. This paper reviews some well known results of Lie group analysis, as well as some recent contributions concerned with the transformation of differential equations to equivalent forms useful to investigate applied problems.

When nonautonomous equations are equivalent to autonomopus ones

We consider nonlinear systems of first order partial differential equations admitting at least two one-parameter Lie groups of transformations with commuting infinitesimal operators. Under suitable conditions it is possible to introduce a variable transformation based on canonical variables which reduces the model in point to autonomous form. Remarkably, the transformed system may admit constant solutions to which there correspond non-constant solutions of the original model. The results are specialized to the case of first order quasilinear systems admitting either dilatation or spiral groups of transformations and a systematic procedure to characterize special exact solutions is given. At the end of the paper the equations of axi-symmetric gas dynamics are considered.

Dynamics and wave propagation in dilatant granular materials

The equations of motion for dilatant granular material are obtained from a Hamiltonian variational principle of local type in the conservative case. The propagation of nonlinear waves in a region with uniform state is studied by means of an asymptotic approach that has already appeared useful in an investigation on wave propagation in bubbly liquids and in fluid mixtures. When the grains are assumed to be incompressible, it is shown that the material behaves as a continuum with latent microstructure.SommarioSi ricavano le equazioni di moto per i materiali granulari dilatanti da un principio variazionale Hamiltoniano di tipo locale nel caso conservativo. Si studia la propagazione delle onde non lineari in una regione di stato costante per mezzo di un approccio asintotico già rivelatosi utile nello studio della propagazione di onde nei liquidi con bolle e nelle miscele di fluidi. Quando si supponga che i granuli siano incomprimibili, si dimostra che il materiale si comporta come un continuo con microstruttura latente.

A phenomenological operator description of interactions between populations with applications to migration

We adopt an operatorial method based on the so-called creation, annihilation and number operators in the description of different systems in which two populations interact and move in a two-dimensional region. In particular, we discuss diffusion processes modeled by a quadratic hamiltonian. This general procedure will be adopted, in particular, in the description of migration phenomena. With respect to our previous analogous results, we use here fermionic operators since they automatically implement an upper bound for the population densities.

How to build up variable transformation allowing one to mapo nonlinear hyperbolic equations into autonomous or linear ones

The paper claims to give a systematic approach allowing one to obtain invertible variable transformations mapping nonlinear partial differential equations either into autonomous or linear form provided that some suitable conditions are satisfed. The procedure makes use of Lie group analysis so that some symmetries are required in order to obtain the required transformations, which are related to canonical variables. A linearization procedure is given in the last part of the paper valid for systems of partial differential equations that can be reduced to a single nonlinear equation linearly degenerate. The procedures are explained with some examples of physical interest.

Reduction to autonomous form by group analysis and exact solutions of axisymmetric MHD equations

Motivated by many physical applications, we consider a general first order system of nonlinear partial differential equations involving two independent variables x, t and a vector field u(x,t). We suppose that the system admits two one parameter Lie groups of transformations with commuting infinitesimal operators. Then, by introducing canonical variables, it is possible to show, under suitable conditions, that the original governing system may be written in autonomous form. Of course, constant solutions of the transformed system are nonconstant solutions of the original system. By using the above mentioned result, we are able to point out a systematic procedure that allows us to build up special nonconstant solutions provided that the governing system admits two commuting infinitesimal operators. The results obtained in [1], in the case when the system under consideration is invariant with respect to the stretching group of transformation can be recovered as a special case of our procedure. In dealing with waves propagating in media having cylindrical or spherical symmetry, as well as in regions with inhomogeneities, we are led, in the one-dimensional case, to consider hyperbolic systems where the independent variables appear explicitly in the coefficients of the system. Consequently, it becomes a suitable procedure to transform the system in autonomous form characterizing, in the meantime, exact solutions. Moreover, it is possible to study discontinuity and shock waves propagating in such special non-constant states as propagation problems in constant states of the transformed system.

An Operator-Like Description of Love Affairs

We adopt the so-called occupation number representation, originally used in quantum mechanics and recently considered in the description of stock markets, in the analysis of the dynamics of love relations. We start with a simple model, involving two actors (Alice and Bob): in the linear case we obtain periodic dynamics, whereas in the nonlinear regime, either periodic or quasi-periodic solutions are found. Then we extend the model to a love triangle involving Alice, Bob, and a third actress, Carla. Interesting features appear, and in particular we find analytical conditions for the linear model of the love triangle to have periodic or quasi-periodic solutions. Numerical solutions are exhibited in the nonlinear case.

Reduction of nonhomogeneous quasilinear 2×2 systems to homogeneous and autonomous form

By using the invariance with respect to suitable Lie groups of point transformations, necessary and sufficient conditions are determined allowing one to map through an invertible point transformation nonhomogeneous and nonautonomous quasilinear 2×2 systems to homogeneous and autonomous form. The procedure is constructive and the new independent and dependent variables are obtained by determining the canonical variables associated with the Lie groups of point symmetries admitted by the source system. Various examples of physical interest arising from different contexts are considered.

A phenomenological operator description of dynamics of crowds: Escape strategies

Abstract We adopt an operatorial method, based on creation, annihilation and number operators, to describe one or two populations mutually interacting and moving in a two-dimensional region. In particular, we discuss how the two populations, contained in a certain two-dimensional region with a non-trivial topology, react when some alarm occurs. We consider the cases of both low and high densities of the populations, and discuss what is changing as the strength of the interaction increases. We also analyze what happens when the region has either a single exit or two ways out.

Nonlinear seismic waves: A model for site effects

Abstract The aim of this paper is to propose a possible mathematical model of site effects that occur when seismic waves propagate through a sediment filled basin. The model is based on the mechanical properties of the medium (that we consider as a granular material) through which the seismic waves propagate. By looking for asymptotic solutions having the features of a progressive wave, we derive an evolution equation which is a modified Korteweg–deVries–Burgers equation containing also a nonlinear dissipative term. This equation is integrated numerically and the modelled site amplification is evaluated by using the smoothed spectral ratio between the propagated profile of the wave and the initial one.