Fourth-order dispersive systems on the one-dimensional torus

The Initial Value Problem for a Third Order Dispersive Equation on the Two-dimensional Torus

|α|=2 2! α! aσ(α)(x)∂ α +~b(x) · ∇+ c(x), ∂t = ∂/∂t, ∂j = ∂/∂xj , ∂ = ∇ = (∂1, ∂2), p(ξ) = ξ1ξ2(ξ1 + ξ2), α = (α1, α2) is a multi-index, |α| = α1 + α2, α! = α1!α2!, σ(α) = (α1 − α2)/2, ~b(x) = (b1(x), b2(x)), and aσ(α), bj(x), c(x) are real-valued smooth functions on T2. Such operators arise in the study of gravity wave of deep water. See [1], [3], [4] and the references therein. The purpose of...

Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition.

We study the distribution of the mean radial displacement of charges of a two-dimensional (2D) one-component plasma in the thermodynamic limit N→∞ at finite temperature β>0. We compute explicitly the large deviation functions showing the emergence of a fourth-order phase transition as a consequence of a change of topology in the plasma distribution. This weak phase transition occurs exactly at ...

Discrete gradient algorithms of high order for one-dimensional systems

We show how to increase the order of one-dimensional discrete gradient numerical integrator without losing its advantages, such as exceptional stability, exact conservation of the energy integral and exact preservation of the trajectories in the phase space. The accuracy of our integrators is higher by several orders of magnitude as compared with the standard discrete gradient scheme (modified ...

The infinite-dimensional torus

Potential Theory on the Infinite Dimensional Torus*

During the last 15 years there has been developped an extensive potential theory for harmonic spaces. The main motivation has been to unify the study of the solutions to various elliptic and parabolic partial differential equations of second order on an open subset of IR" or more generally on a differentiable manifold. Classical potential theory in IR" is translation invariant and it is therefo...