In this paper, we further develop recent results in the numerical solution of Hamiltonian partial differential equations (PDEs) [14], by means of energyconserving methods in the class of Line Integral Methods, in particular, the Runge-Kutta methods named Hamiltonian Boundary Value Methods (HB-VMs). We shall use HBVMs for solving the nonlinear Schrödinger equation (NLSE), of interest in many applications. We show that the use of energyconserving methods, able to conserve a discrete counterpart of the Hamiltonian functional, confers more robustness on the numerical solution of such a problem.

We consider a two-dimensional electron gas with a spin-orbit interaction of Bychkov and Rashba type. Starting from a microscopic model, represented by the von Neumann equation endowed with a suitable Bhatnagar–Gross–Krook collision term, we apply the Chapman–Enskog method to derive a quantum diffusive model. Such model is then semiclassically expanded up to second order, leading to nonlinear quantum corrections to the zero-order diffusive models of the literature.

The maximum entropy principle is applied to the formal derivation of isothermal, Euler-like equations for semiclassical fermions (electrons and holes) in graphene. After proving general mathematical properties of the equations so obtained, their asymptotic form corresponding to significant physical regimes is investigated. In particular, the diffusive regime, the Maxwell-Boltzmann regime (high temperature), the collimation regime and the degenerate gas limit (vanishing temperature) are considered. C 2014 AIP Publishing LLC. [http://dx.

Many coordination phenomena are based on a synchronisation process, whose global behaviour emerges from the interactions among the individual parts. Often in Nature, such self-organising mechanism allows the system to behave as a whole and thus grounding its very first existence, or expected functioning, on such process. There are however cases where synchronisation acts against the stability of the system; for instance in the case of engineered structures, resonances among sub parts can destabilise the whole system. In this Letter we propose an innovative control method to tackle the synchronisation process based on the use of the Hamiltonian control theory, by adding a small control term to the system we are able to impede the onset of the synchronisation. We present our results on the paradigmatic Kuramoto model but the applicability domain is far more large.PACS numbers: 05.45. Xt, 02.30.Yy, 45.20.Jj Synchronisation is one of the most important example of collective behaviour in Nature, being at the basis of many processes in living beings [1][2][3]. Indeed, their activity is regulated by (almost) periodic processes of different duration that must run at unison to determine a collective behaviour [4,5] enable to sustain life. For this reason synchronisation is widespread and it has been studied in many research domains such as biology (flashing of fireflies [6] and the cricket chirping [7] during the mating season), chemistry (glycolytic oscillations in populations of yeast cells [8]), physics (arrays of coupled lasers [9] and the superconducting Josephson junctions [10]), just to mention few of them. One of the most representative case being the heart [11], an organ of vital importance for all species in the animal kingdom; the heart is composed by a collection of individual cells, myocytes, whose complex interactions among them are responsible for the ability to pump blood in the circulatory system. Initially, at embryonic state, such cells do not interact each other and their beats are independent, only after a couple of days, the myocytes form interconnected sheets of cells that help them beat in unison. The absence of such synchronisation phenomenon in human induces cardiac arrhythmia and artificial pacemakers are necessary to recover the normal behaviour.Despite the very different nature of the systems exhibiting synchronisation phenomena, most of the main features are quite universal and can thus be described using the paradigmatic Kuramoto model (KM) [12][13][14][15] of coupled non-linear oscillators. Once the N oscillators are set on top of a complex network [16], the KM can be described byφwhere K is the interaction strength, ω i are the natural frequencies of the oscillators drawn from some distribu- Let us observe that there are cases where such collective rhythm has a negative impact on the organism life [17]; for instance, it has been observed, that certain psychomotor symptoms, e.g. tremors, are results of an abnormal synchronisation phenomenon in the activity of the responsible neuronal zo...

In this paper the effective mass approximation and k·p multi-band models, describing quantum evolution of electrons in a crystal lattice, are discussed. Electrons are assumed to move in both a periodic potential and a macroscopic one. The typical period ǫ of the periodic potential is assumed to be very small, while the macroscopic potential acts on a much bigger length scale. Such homogenization asymptotic is investigated by using the envelope-function decomposition of the electron wave function. If the external potential is smooth enough, the k·p and effective mass models, well known in solid-state physics, are proved to be close (in strong sense) to the exact dynamics. Moreover, the position density of the electrons is proved to converge weakly to its effective mass approximation.