Static- and dynamical-phase transition in multidimensional voting models on continua

A voting model (or a generalization of the Glauber model at zero temperature) on a multidimensional lattice is defined as a system composed of a lattice each site of which is either empty or occupied by a single particle. The reactions of the system are such that two adjacent sites, one empty the other occupied, may evolve to a state where both of these sites are either empty or occupied. The continuum version of this model in a Ddimensional region with boundary is studied, and two general behaviors of such systems are investigated. The stationary behavior of the system, and the dominant way of the relaxation of the system toward its stationary state. Based on the first behavior, the static phase transition (discontinuous changes in the stationary profiles of the system) is studied. Based on the second behavior, the dynamical phase transition (discontinuous changes in the relaxation-times of the system) is studied. It is shown that the static phase transition is induced by the bulk reactions only, while the dynamical phase transition is a result of both bulk reactions and boundary conditions.Comment: 10 pages, LaTeX2

New conformal mapping for adaptive resolving of the complex singularities of Stokes wave

A new highly efficient method is developed for computation of traveling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularites above the free surface for the analytical continuation of the travelling wave into the complex plane. An auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics. The efficiency of that conformal mapping is demonstrated for Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows to provide a detailed study of the oscillatory approach of these solutions to the limiting wave. Generalizations of the conformal mapping to resolve multiple singularities are also introduced

Branch cuts of Stokes wave on deep water. Part I: Numerical solution and Pad\'e approximation

Complex analytical structure of Stokes wave for two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth is analyzed. Stokes wave is the fully nonlinear periodic gravity wave propagating with the constant velocity. Simulations with the quadruple and variable precisions are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with $2\pi/3$ radians angle on the crest. A conformal map is used which maps a free fluid surface of Stokes wave into the real line with fluid domain mapped into the lower complex half-plane. The Stokes wave is fully characterized by the complex singularities in the upper complex half-plane. These singularities are addressed by rational (Pad\'e) interpolation of Stokes wave in the complex plane. Convergence of Pad\'e approximation to the density of complex poles with the increase of the numerical precision and subsequent increase of the number of approximating poles reveals that the only singularities of Stokes wave are branch points connected by branch cuts. The converging densities are the jumps across the branch cuts. There is one branch cut per horizontal spatial period $\lambda$ of Stokes wave. Each branch cut extends strictly vertically above the corresponding crest of Stokes wave up to complex infinity. The lower end of branch cut is the square-root branch point located at the distance $v_c$ from the real line corresponding to the fluid surface in conformal variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Pad\'e approximation for Stokes waves of different heights are provided. These tables allow to recover the Stokes wave with the relative accuracy of at least $10^{-26}$. The tables use from several poles to about hundred poles for highly nonlinear Stokes wave with $v_c/\lambda\sim 10^{-6}.$Comment: 38 pages, 9 figures, 4 tables, supplementary material