In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group Ǧ, contructible with respect to the stratification by Ǧ(C[[x]])-orbits. Following ideas of Ginzburg and Arkhipov– Bezrukavnikov–Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of the Langlands dual reductive group G. We also ...

The moduli spaces of two (0, 2) compactifications of the heterotic string can share the same Landau-Ginzburg model even though at large radius they look completely different. It was argued that such a pair of (0, 2) models might be connected via a perturbative transition at the Landau-Ginzburg point. Situations of this kind are studied for some explicit models. By calculating the exact dimensio...

By means of the Ginzburg-Landau theory of phase transitions, we study a non-isothermal model to characterize the austenite-martensite transition in shape memory alloys. In the first part of this paper, the onedimensional model proposed in [3] is modified by varying the expression of the free energy. In this way, the description of the phenomenon of hysteresis, typical of these materials, is imp...

We introduce and discuss discrete two-dimensional models for XY spin systems and screw dislocations in crystals. We prove that, as the lattice spacing ε tends to zero, the relevant energies in these models behave like a free energy in the complex Ginzburg-Landau theory of superconductivity, justifying in a rigorous mathematical language the analogies between screw dislocations in crystals and v...

We study, analytically, the discrete complex cubic Ginzburg–Landau (dCCGL) equation. We derive the energy balance equation for the dCCGL and consider various limiting cases. We have found a set of exact solutions which includes as particular cases periodic solutions in terms of elliptic Jacobi functions, bright and dark soliton solutions, and constant magnitude solutions with phase shifts. We h...

The Ginzburg–Landau equation has been used as a mathematical model for various pattern formation systems in mechanics, physics and chemistry. In this paper, we study the complex Ginzburg–Landau equation in two spatial dimensions with periodical boundary conditions. The method numerically approximates the solution by collocation method based on radial basis functions (RBFs). To improve the numer...

We study the topological heterotic ring in (0,2) Landau-Ginzburg models without a (2,2) locus. The ring elements correspond to elements of the Koszul cohomology groups associated to a zero-dimensional ideal in a polynomial ring, and the computation of half-twisted genus zero correlators reduces to a map from the first nontrivial Koszul cohomology group to complex numbers. This map is a generali...

We study a parametrically driven Ginzburg-Landau equation model with nonlinear management. The system is made of laterally coupled long active waveguides placed along a circumference. Stationary solutions of three kinds are found: periodic Ising states and two types of Bloch states, staggered and unstaggered. The stability of these states is investigated analytically and numerically. The nonlin...