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 (32 digits) and variable precisions (more than 200 digits) are performed to find Stokes wave with high accuracy and study the Stokes wave approaching its limiting form with 2π/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 (Padé) interpolation of Stokes wave in the complex plane. Convergence of Padé 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 λ 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 increase of the scaled wave height H/λ from the linear limit H/λ = 0 to the critical value H max /λ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave (also called by the Stokes wave of the greatest height). Here H is the wave height from the crest to the trough in physical variables. The limiting Stokes wave emerges as the singularity reaches the fluid surface. Tables of Padé 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 for near-linear Stokes wave up to about hundred poles to highly nonlinear Stokes wave with v c /λ ∼ 10 −6 .

The dynamics of Bose-Einstein condensates of a gas of bosonic particles with long-range dipole-dipole interactions in a harmonic trap is studied. Sufficient analytical criteria are found both for catastrophic collapse of Bose-Einstein condensates and for long-time condensate existence. Analytical criteria are compared with variational analysis.

A connection is established between discrete stochastic model describing microscopic motion of fluctuating cells, and macroscopic equations describing dynamics of cellular density. Cells move towards chemical gradient (process called chemotaxis) with their shapes randomly fluctuating. Nonlinear diffusion equation is derived from microscopic dynamics in dimensions one and two using excluded volume approach. Nonlinear diffusion coefficient depends on cellular volume fraction and it is demonstrated to prevent collapse of cellular density. A very good agreement is shown between Monte Carlo simulations of the microscopic cellular Potts model and numerical solutions of the macroscopic equations for relatively large cellular volume fractions. Combination of microscopic and macroscopic models were used to simulate growth of structures similar to early vascular networks.

A new highly efficient method is developed for computation of travelling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularities 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 the Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows us 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.

Building on the scientific understanding and technological infrastructure of single-mode fibers, multimode fibers are being explored as a means of adding new degrees of freedom to optical technologies such as telecommunications, fiber lasers, imaging, and measurement. Here, starting from a baseline of single-mode nonlinear fiber optics, we introduce the growing topic of multimode nonlinear fiber optics. We demonstrate a new numerical solution method for the system of equations that describes nonlinear multimode propagation, the generalized multimode nonlinear Schrödinger equation. This numerical solver is freely available, implemented in MATLAB ® and includes a number of multimode fiber analysis tools. It features a significant parallel computing speed-up on modern graphical processing units, translating to orders-of-magnitude speed-up over the split-step Fourier method. We demonstrate its use with several examples in graded-and step-index multimode fibers. Finally, we discuss several key open directions and questions, whose answers could have significant scientific and technological impact.

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In the development of multiscale biological models it is crucial to establish a connection between discrete microscopic or mesoscopic stochastic models and macroscopic continuous descriptions based on cellular density. In this paper a continuous limit of a two-dimensional Cellular Potts Model (CPM) with excluded volume is derived, describing cells moving in a medium and reacting to each other through both direct contact and long range chemotaxis. The continuous macroscopic model is obtained as a Fokker-Planck equation describing evolution of the cell probability density function. All coefficients of the general macroscopic model are derived from parameters of the CPM and a very good agreement is demonstrated between CPM Monte Carlo simulations and numerical solution of the macroscopic model. It is also shown that in the absence of contact cell-cell interactions, the obtained model reduces to the classical macroscopic Keller-Segel model. General multiscale approach is demonstrated by simulating spongy bone formation from loosely packed mesenchyme via the intramembranous route suggesting that self-organizing physical mechanisms can account for this developmental process. A large literature exists studying continuous limits of point-wise discrete microscopic models for biological systems. For example, the classic Keller-Segel PDE model of chemotaxis [1] was derived from a discrete model with point-wise cells undergoing random walk [2][3][4][5]. However, many biological phenomena require taking into account the finite size of biological cells, and much less work has been done on deriving macroscopic limits of microscopic models which treat cells as extended objects. The mesoscopic Cellular Potts Model (CPM), first introduced by Glazier and Graner [6,7], has been used as a component of multiscale, experimentally motivated hybrid approaches, combining discrete and macroscopic continuous representations, to simulate, among others, morphological phenomena in the cellular slime mold Dictyostelium discoideum [8], vascular development [9] and the proximo-distal increase in the number of skeletal elements in the developing avian limb [10].One of the earliest attempts at combining mesoscopic and macroscopic levels of description of cellular dynamics was described in [11] where the diffusion coefficient for a collection of noninteracting randomly moving cells was derived from a one-dimensional CPM. Recently a microscopic limit of subcellular elements model [12] was derived in the form of an advection-diffusion partial differential equation for cellular density. In previous papers [13,14] we studied the continuous limit of 1D and 2D models of individual cell motion in a medium, in the presence of an external field but without contact cell-cell interactions.This paper describes a theoretical analysis leading to a continuous macroscopic limit of the two-dimensional

The cellular Potts model (CPM) has been used for simulating various biological phenomena such as differential adhesion, fruiting body formation of the slime mold Dictyostelium discoideum, angiogenesis, cancer invasion, chondrogenesis in embryonic vertebrate limbs, and many others. We derive a continuous limit of a discrete one-dimensional CPM with the chemotactic interactions between cells in the form of a Fokker-Planck equation for the evolution of the cell probability density function. This equation is then reduced to the classical macroscopic Keller-Segel model. In particular, all coefficients of the Keller-Segel model are obtained from parameters of the CPM. Theoretical results are verified numerically by comparing Monte Carlo simulations for the CPM with numerics for the Keller-Segel model.