Asymptotic Methods in Nonlinear Wave Phenomena

PREFACE.

Conference Committees.

CONTENTS.

We consider a variational formulation of the steady Boltzmann equation for semiconductors. We apply this formulation to the calculation of an approximate expression of the electron mobility, valid for electric fields up to 1-1.5 .

In 1999 Chua demonstrated that it is possible to obtain boolean Cellular Automata (CA) by using Cellular Neural Networks (CNNs), implemented as chips, which definitely reduce the time of simulation for discrete dynamical systems such as CA and allow for a better understanding of complexity. In this work we demonstrate that Chua's Universal Neuron which simulates boolean CA can be generalized for multistate CA. This new approach allows to investigate the nature of the CA rule space, in relationship with the Universal Neuron parameter space, establishing relationships between discrete and continuous dynamical systems and analysing the nature of local and global rules in affecting the behaviour of these systems. The method which we have developed is fruitful since we have found a lot of CA which can be considered complex rules of class IV, in the Wolfram classification. Furthermore we have used the idea of genetical computation in considering the values of the Universal Neuron parameter space as a sort of genotype which determines and produces variations in the CA behaviour. On this basis, we have implemented a genetic algorithm which fits very well with the searching for CA complex rules.

The nature of the relationship between continuous and discrete systems is a very important topic in the Science of Complexity which helps us to clarify the temporal dimension of many biological phenomena and processes.

The equations of isocline curves can be obtained from a variational principle with the exceptional scalar field Lagrangian. Shocks are shown to propagate on characteristic curves. Monge-Ampère equation, von Kármán fluid, and Born-Infeld theory appear as examples.

We discuss the interplay between Fokker-Planck dynamics and some elementary aspects of the theory of Hamilton-Perelman Ricci flow. In particular, we show that the Fokker-Planck equation is the natural diffusion process for absolutely continuous probability measures (backward) evolving along the Ricci flow. A few preliminary results on the asymptotic trend to equilibrium for such a process are reported.

A model equation which describes reaction diffusion processes is considered. We classify the possible forms of the material response functions involved in the governing equation under interest allowing generalized separable exact solutions to hold. Exact solutions of a nonlinear second order reaction diffusion equation are calculated.

The general linear response theory it applied for study mechanical relaxations phenomena in continuous media. It is shown that phenomenological and state coefficients, in viscoanelastic media can be obtained as function of quantities which are experimentally measurable. The behaviour of the medium at low and high frequency is taking in account. Therefore, by compared the solutions of the differential equations that describe relaxation phenomena in the considered model to a relation that describe the same phenomena in linear response theory, it is possible to determine the aforementioned coefficients.

The sixth Painlevé equation P6 still lacks a second order matrix Lax pair which would be holomorphic in the four parameters of P6. We investigate here the construction of such a Lax pair from the reduction to P6 of some physical integrable system depending on space and time. Choosing this system as the three-wave resonant interaction, the reduction found by Kitaev yields a third order matrix Lax pair with explicit entries in terms of P6. In order to lower the matrix order to two, two methods can be used. The one due to Harnad reduces to the question of factorizing a rank two third order matrix into the product of two rectangular matrices.

Motions of continuous media presenting singularities are associated with phenomena involving shocks, interfaces or material surfaces. The equations representing evolutions of these media are irregular through geometrical manifolds. A unique continuous medium is conceptually simpler than several media with surfaces of singularity. To avoid the surfaces of discontinuity in the theory, we transform the model by considering a continuous medium taking into account more complete internal energies expressed in gradient developments associated with the variables of state. Nevertheless, resulting equations of motion are of an higher order than those of the classical models: they lead to non-linear models associated with more complex integration processes on the mathematical level as well as on the numerical point of view. In fact, such models allow a precise study of singular zones when they have a non negligible physical thickness. This is typically the case for capillarity phenomena in fluids or mixtures of fluids in which interfacial zones are transition layers between phases or layers between fluids and solid walls. Within the framework of mechanics for continuous media, we propose to deal with the functional point of view considering globally the equations of the media as well as the boundary conditions associated with these equations. For this aim, we revisit the d'Alembert-Lagrange principle of virtual works which is able to consider the expressions of the works of forces applied to a continuous medium as a linear functional value on a space of test functions in the form of virtual displacements. At the end, we analyze examples corresponding to capillary fluids. This analysis brings us to numerical or asymptotic methods avoiding the difficulties due to singularities in simpler -but with singularities- models.

A dynamical system representing the time evolution of bioenergy in an environmental system is here proposed. After the derivation of the model and a detailed description of the parameters, the equilibria of the dynamical system are detected and the relative stability analysis is performed. Finally an application to a real case is shown.

The canonical reduction method,^1–3 is applied to studying the stability of the zero solution of the Lorenz system with and without rotation. Some stability results of the motionless state of the rotating Bénard problem are also given.

Within the context of the inverse Lie problem the question whether there exist PDEs that are characterized by their Lie point symmetries may be addressed. In a recent paper the authors called these equations Lie remarkable. In this paper we exhibit various examples of Lie remarkable equations, including some multidimensional Monge-Ampère type equations.

Pulse and shock interaction is investigated for a class of ideally hard elastic materials. The latter admit the soliton-like property that disturbances emerge asymptotically unaltered in shape and speed following the interaction process.

No abstract received.

This is part of a talk given by the author at the conference held in Palermo (2006) in honor of Prof. Antonio Greco.

No abstract received.

In this paper a hyperbolic model is proposed for mixtures of gases which are neither viscous, nor heat-conducting (Eulerian fluids). It is built upon assumption that each constituent obeys it's own temperature. Restrictions to the structure of the model come out from basic principles of extended thermodynamics, i.e. Galilean invariance of balance laws and entropy inequality. Hierarchy of hyperbolic subsystems is recognized, with a single-temperature model as principal subsystem and classical Euler's equations as equilibrium subsystem. Finally, in order to relate this model to classical thermodynamics, a Maxwellian iteration is performed in the case of binary mixture, giving rise to a relation between the difference of non-equilibrium temperatures of constituents and classical fields.

The sequence of equilibria of two dimensional reduced magnetohydrodynamics has been studied as a function of the aspect ratio ∊ as a control parameter of the magnetic shear for different equilibrium magnetic fields. A threshold in the magnetic shear parameter is found, below which the small island reconnected equilibrium disappears by tangent bifurcation, while above it undergoes a further symmetry-breaking bifurcation leading to equilibria with enhanced kinetic energy in the form of zonal flow. The spontaneous development of zonal flows may ameliorate transport and improve confinement in tokamak devices.

We get equivalence transformations for a class of reaction-diffusion systems. After having specialized them for a family of models concerned with bacterial colonies, we derive symmetries and some invariant solutions.