Interaction of Ising-Bloch fronts with Dirichlet Boundaries

We study the Ising-Bloch bifurcation in two systems, the Complex Ginzburg Landau equation (CGLE) and a FitzHugh Nagumo (FN) model in the presence of spatial inhomogeneity introduced by Dirichlet boundary conditions. It is seen that the interaction of fronts with boundaries is similar in both systems, establishing the generality of the Ising-Bloch bifurcation. We derive reduced dynamical equations for the FN model that explain front dynamics close to the boundary. We find that front dynamics in a highly non-adiabatic (slow front) limit is controlled by fixed points of the reduced dynamical equations, that occur close to the boundary.Comment: 10 pages, 8 figures, submitted to Phys. Rev.

Ground state phase diagram of a spinless, extended Falicov-Kimball model on the triangular lattice

Correlated systems with hexagonal layered structures have come to fore with renewed interest in Cobaltates, transition-metal dichalcogenides and GdI2. While superconductivity, unusual metal and possible exotic states (prevented from long range order by strong local fluctuations) appear to come from frustration and correlation working in tandem in such systems, they freeze at lower temperature to crystalline states. The underlying effective Hamiltonian in some of these systems is believed to be the Falicov-Kimball model and therefore, a thorough study of the ground state of this model and its extended version on a non-bipartite lattice is important. Using a Monte Carlo search algorithm, we identify a large number of different possible ground states with charge order as well as valence and metal-insulator transitions. Such competing states, close in energy, give rise to the complex charge order and other broken symmetry structures as well as phase segregations observed in the ground state of these systems.Comment: 9 pages, 7 figure

An hp-Local Discontinuous Galerkin method for Parabolic\ud Integro-Differential Equations

In this article, a priori error analysis is discussed for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that the L2 -norm of the gradient and the L2 -norm of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains