Dark topological valley Hall edge solitons

Topological edge solitons propagating along the edge of a photonic topological insulator are localized self-sustained hybrid states that are immune to defects/disorders due to the protection of the edge states stemming from the nontrivial topology of the system. Here, we predict that exceptionally robust dark valley Hall edge solitons may form at the domain walls between two honeycomb lattices with broken inversion symmetry. The underlying structure can be created with femtosecond laser inscription, it possesses a large bandgap where well-localized dark edge solitons form, and in contrast to systems with broken time-reversal symmetry, it does not require external magnetic fields or complex longitudinal waveguide modulations for the realization of the topological phase. We present the envelope equation allowing constructing dark valley Hall edge solitons analytically. Such solitons propagate without radiation into the bulk of the lattice and can circumvent sharp corners, which allows observing their persistent circulation along the closed triangular domain wall boundary. They survive over huge distances even in the presence of disorder in the underlying lattice. We also investigate interactions of closely located dark topological valley Hall edge solitons and show that they are repulsive and lead to the formation of two gray edge solitons, moving with different group velocities departing from group velocity of the linear edge state on which initial dark solitons were constructed. Our results illustrate that nonlinear valley Hall systems can support a rich variety of new self-sustained topological states and may inspire their investigation in other nonlinear systems, such as atomic vapors and polariton condensates.

Bijections Between Walks Inside a Triangular Domain and Motzkin Paths of Bounded Amplitude

This paper solves an open question of Mortimer and Prellberg asking for an explicit bijection between two families of walks. The first family is formed by what we name triangular walks, which are two-dimensional walks moving in six directions (0°, 60°, 120°, 180°, 240°, 300°) and confined within a triangle. The other family is comprised of two-colored Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height. We provide several new bijections. The first one is derived from a simple inductive proof, taking advantage of a 2n-to-one function from generic triangular walks to triangular walks only using directions 0°, 120°, 240°. The second is based on an extension of Mortimer and Prellberg's results to triangular walks starting not only at a corner of the triangle, but at any point inside it. It has a linear-time complexity and is in fact adjustable: by changing some set of parameters called a scaffolding, we obtain a wide range of different bijections. Finally, we extend our results to higher dimensions. In particular, by adapting the previous proofs, we discover an unexpected bijection between three-dimensional walks in a pyramid and two-dimensional simple walks confined in a bounded domain shaped like a waffle.

A New Quasi Cubic Rational System with Two Parameters

The purpose of this article is to develop a new system for the construction of curves and surfaces. Making the new system not only has excellent properties of the orthodox Bézier and the B-spline method but also has practical properties such as variation diminishing and local shape adjustability. First, a new set of the quasi-cubic rational (QCR) system with two parameters is given, which is verified on an optimal normalized totally positive system (B-system). The related QCR Bézier curve is defined, and the de Casteljau-type algorithm are given. Next, a group of non-uniform QCR B-spline system is shown based on the linear combination of the proposed QCR system, the relative properties of the B-spline system are analyzed. Then, the definition and properties of non-uniform QCR B-spline curves are discussed in detail. Finally, the proposed QCR system is extended to the triangular domain, which is called the quasi-cubic rational Bernstein-Bézier (QCR-BB) system, and its related definition and properties of patches are given at length. The experimental image obtained by using MATLAB shows that the newly constructed system has excellent properties such as symmetry, totally positive, and C 2 continuity, and its corresponding curve has the properties of local shape adjustability and C 2 continuity. These extended systems in the extended triangular domain have symmetry, linear independence, etc. Hence, the methods in this article are suitable for the modeling design of complex curves and surfaces.

Image Zooming Based on Two Classes of C1-Continuous Coons Patches Construction with Shape Parameters over Triangular Domain

Image interpolation is important in image zooming. To improve the quality of image zooming, in this work, we proposed a class of rational quadratic trigonometric Hermite functions with two shape parameters and two classes of C 1 -continuous Coons patches constructions over a triangular domain by improved side–side method and side–vertex method. Altering the values of shape parameters can adjust the interior shape of the triangular Coons patch without influencing the function values and partial derivatives of the boundaries. In order to deal with the problem of well-posedness in image zooming, we discussed symmetrical sufficient conditions for region control of shape parameters in the improved side–side method and side–vertex method. Some examples demonstrate the proposed methods are effective in surface design and digital image zooming. C 1 -continuous Coons patches constructed by the proposed methods can interpolate to scattered 3D data. By up-sampling to the constructed interpolation surface, high-resolution images can be obtained. Image zooming experiment and analysis show that compared to bilinear, bicubic, iterative curvature-based interpolation (ICBI), novel edge orientation adaptive interpolation scheme for resolution enhancement of still images (NEDI), super-resolution using iterative Wiener filter based on nonlocal means (SR-NLM) and rational ball cubic B-spline (RBC), the proposed method can improve peak signal to noise ratio (PSNR) and structural similarity index (SSIM). Edge detection using Prewitt operator shows that the proposed method can better preserve sharp edges and textures in image zooming. The proposed methods can also improve the visual effect of the image, therefore it is efficient in computation for image zooming.

The hyperbolic Ernst equation in a triangular domain

The collision of two plane gravitational waves in Einstein’s theory of relativity can be described mathematically by a Goursat problem for the hyperbolic Ernst equation in a triangular domain. We use the integrable structure of the Ernst equation to present the solution of this problem via the solution of a Riemann–Hilbert problem. The formulation of the Riemann–Hilbert problem involves only the prescribed boundary data, thus the solution is as effective as the solution of a pure initial value problem via the inverse scattering transform. Our results are valid also for boundary data whose derivatives are unbounded at the triangle’s corners—this level of generality is crucial for the application to colliding gravitational waves. Remarkably, for data with a singular behavior of the form relevant for gravitational waves, it turns out that the singular integral operator underlying the Riemann–Hilbert formalism can be explicitly inverted at the boundary. In this way, we are able to show exactly how the behavior of the given data at the origin transfers into a singular behavior of the solution near the boundary.