Nonlinear dynamics in spatially extended systems

We address coarse-graining issues arising from nonlinear systems in nonequilibrium states. First, we investigate the so-called renormalization group method and its application to the kinetic theories. Using the renormalization (RG) group method we show that the Fokker-Planck equation and the generalized Kramers-Moyal expansion are RG equations. It is shown that the only Gaussian noise that renders the Fokker-Planck equation valid for all orders, in terms of renormalizability, is a white one. For non-Gaussian noises the Kramers-Moyal expansion appears naturally from the RG method under the same requirement of renormalizability to all orders.; Next we apply the same renormalization group method to derive the kinetic equation of weak turbulence. The derivation of the kinetic equation of nonlinear dispersive waves involves coarse-graining in time and is similar to the approximation used to derive the Boltzmann equation via the BBGKY hierarchy. In the same manner as the Boltzmann equation describes the evolution of the gas system in nonequilibrium for macroscopic times, the kinetic equation of weak turbulence describes nonequilibrium states of weakly interacting waves over coarse-grained macroscopic times.; Furthermore, we turn our attention to the spatiotemporal structures related to biological phenomena of bacterial motion. Experimentally, it is observed that jets and swirls of bacterial motion appear in a solution if the concentration of bacterial microorganisms is sufficiently high. A simple model of self-propelled particles is proposed to show that coherent structures, such as jets and swirls, can arise from plausible microscopic mechanisms: (i) the elongated shape of the self-propelled particles with (ii) the hardcore interactions among them. We demonstrate via computer simulation that these coherent structures, which emerge at sufficiently high densities of particles, have characteristics that are similar to those observed in recent experiments in bacterial baths.; Finally, we study the spatiotemporal patterns that emerge from neuronal networks with an architecture similar to that of the Watz-Strogatts network. It is shown that under certain conditions the neural network system exhibits oscillations. These oscillations exhibit a wide range of frequencies in the Gamma band.