This work introduces a new operator-splitting method for solving the two-dimensional (2D) and three-dimensional (3D) parabolic equations. The aim is to reduce the computational complexity without loss of accuracy. Firstly, we split the 2D and 3D parabolic equations into a sequence of one-dimensional (1D) parabolic equations respectively, then we solve each 1D parabolic equation by using finite ...

Following Parreau’s work in 1951-52, we give a unified definition of parabolic Riemann surfaces, with or without boundary. A surface is parabolic under the unified definition implies that it is either relative parabolic or parabolic under the classical definitions. Then we study the conformal structures of noncompact, proper, branched minimal surfaces in R and prove several criteria of such sur...

In this paper, a nonlinear inverse problem of parabolic type, is considered. By reducing this inverse problem to a system of Volterra integral equations the existence, uniqueness, and stability of the solution will be shown.

The concept of the iterative parabolic approximation based on the multiscale technique is discussed. This approach is compared with the traditional ways to derive the wide-angle parabolic equation. While the latter fail in the nonlinear case, the multiscale derivation technique leading to iterative parabolic equations can be easily adapted to handle it. The nonlinear iterative parabolic approxi...

This paper concerns the moduli spaces of rank two parabolic Higgs bundles and parabolic K(D) pairs over a smooth curve. Precisely which parabolic bundles occur in stable K(D) pairs and stable Higgs bundles is determined. Using Morse theory, the moduli space of parabolic Higgs bundles is shown to be a noncompact, connected, simply connected manifold, and a computation of its Poincaré polynomial ...

We address the dynamics of fundamental and higher-order dark solitons on the intense parabolic background in a planar graded-index waveguide with self-defocusing nonlinearity. It is shown that dark solitons on the parabolic background can propagate stably, and feature the properties similar to dark solitons for standard nonlinear Schrödinger equation. We present an approximate solution for dark...

A surface in hyperbolic space H 3 invariant by a group of parabolic isometries is called a parabolic surface. In this paper we investigate parabolic surfaces of H 3 that satisfy a linear Weingarten relation of the form aκ1 + bκ2 = c or aH + bK = c, where a, b, c ∈ R and, as usual, κi are the principal curvatures, H is the mean curvature and K is de Gaussian curvature. We classify all parabolic ...

The mapping class group of a d-pointed Riemann surface has a natural C∞ action on any moduli space of parabolic bundles with the marked points as the parabolic points. We prove that under some numerical conditions on the parabolic data, the induced action of the mapping class group on the cohomology algebra of the moduli space of parabolic bundles factors through the natural epimorphism of the ...

Parabolic dish solar concentrators are very expensive devices with a cost of up to a half of the total cost of a solar power station. The specific technology of parabolic mirror manufacturing makes the decrease of the cost very problematic. Even in mass production the cost of a parabolic mirror is estimated as $500 per square meter. There is another way to make a parabolic concentrator by appro...

This paper is concerned with the numerical solution of parabolic equations coupled to gradient equations. The gradient equations are ordinary diierential equations whose solutions deene positions of particles in the spatial domain of the parabolic equations. The vector eld of the gradient equations is determined by gradients of solutions to the parabolic equations. Such mixed parabolic-gradient...