Non-radially symmetric solutions to the Ginzburg-Landau equation

We study an atom with finitely many energy levels in contact with a heat bath consisting of photons (black body radiation) at a temperature $T >0$. The dynamics of this system is described by a Liouville operator, or thermal Hamiltonian, which is the sum of an atomic Liouville operator, of a Liouville operator describing the dynamics of a free, massless Bose field, and a local operator describing the interactions between the atom and the heat bath. We show that an arbitrary initial state which is normal with respect to the equilibrium state of the uncoupled system at temperature $T$ converges to an equilibrium state of the coupled system at the same temperature, as time tends to $+ \infty

On collapse of wave maps

We derive the universal collapse law of degree 1 equivariant wave maps (solutions of the sigma-model) from the 2+1 Minkowski space-time,to the 2-sphere. To this end we introduce a nonlinear transformation from original variables to blowup ones. Our formal derivations are confirmed by numerical simulations.Comment: 1 figur

Collapse of an Instanton

We construct a two parameter family of collapsing solutions to the 4+1 Yang-Mills equations and derive the dynamical law of the collapse. Our arguments indicate that this family of solutions is stable. The latter fact is also supported by numerical simulations.Comment: 17 pages, 1 figur

Ginzburg-Landau Equation I. Static Vortices

We consider radially symmetric solutions of the Ginzburg-Landau equation (without magnetic field) in dimension 2. Such solutions are called vortices and are specified by their winding number at infinity (vorticity). For a given vorticity n we prove existence and uniqueness (modulo symmetry transformations) of an n-vortex and show that for n = 0; \Sigma1 such vortices are stable while for jnj 2, unstable. We introduce the renormalized Ginzburg-Landau energy and use it for the existence and uniqueness proof. Our stability proof is novel and uses the concept of symmetry breaking and its consequence in the form of zero modes of the linearized equation

The Ginzburg-Landau Equation II. The Energy Of Vortex Configurations

We consider the Ginzburg-Landau equation in dimension two. We introduce a key notion of the energy of vortex configurations. It is defined by minimizing the renormalized Ginzburg-Landau (free) energy introduced in the previous paper over functions with a given set of zeros of given local indices. This notion allows us to define the vortex interaction and vortex Hamiltonian in a canonical way. We find asymptotic behaviour of this energy as the intervortex distances grow. To this end we use several novel techniques