Geometric properties of caustics of pseudo-spherical surfaces

We prove that the nonlinear Schrodinger equation of attractive type (NLS+) describes just spherical surfaces (SS) and the nonlinear Schrodinger equation of repulsive type (NLS-) determines only pseudospherical surfaces (PSS). This implies that, though we show that given two differential PSS (resp. SS) equations there exists a local gauge transformation (despite of changing the independent variables or not) which transforms a solution of one into any solution of the other, it is impossible to have such a gauge transformation between the NLS+ and the NLS-.

Line congruences as surfaces in the space of lines

Conservation laws for some nonlinear evolution equations which describe pseudo-spherical surfaces

We consider evolution equations, mainly of type ut = F(u, ux,..., ∂ku/∂xk), which describe pseudo‐spherical surfaces. We obtain a systematic procedure to determine a linear problem for which a given equation is the integrability condition. Moreover, we investigate how the geometrical properties of surfaces provide analytic information for such equations.

I show that the compound modified Korteweg-de Vries-Sine-Gordon equations describe pseudospherical surfaces, that is, these equations are the integrability conditions for the structural equations of such surfaces. I obtain the self-Bäcklund transformations for these equations by a geometrical method and apply the Bäcklund transformations to these solutions and generate new traveling wave solutions. Conservation laws for the latter ones are obtained using a geometrical property of these pseudospherical surfaces.

Along cuspidal edge singularities on a given surface in Euclidean 3-space $\boldsymbol{R}^3$, which can be parametrized by a regular space curve $\hat\gamma(t)$, a unit normal vector field $\nu$ is well-defined as a smooth vector field of the surface. A cuspidal edge singular point is called generic if the osculating plane of $\hat\gamma(t)$ is not orthogonal to $\nu$. This genericity is equivalent to the condition that its limiting normal curvature $\kappa_\nu$ takes a non-zero value. In this paper, we show that a given generic (real analytic) cuspidal edge $f$ can be isometrically deformed preserving $\kappa_\nu$ into a cuspidal edge whose singular set lies in a plane. Such a limiting cuspidal edge is uniquely determined from the initial germ of the cuspidal edge.

We study the geometry of cuspidal $S_k$ singularities in $\mathbb R^3$ obtained by folding generically a cuspidal edge. In particular we study the geometry of the cuspidal cross-cap $M$, i.e. the cuspidal $S_0$ singularity. We study geometrical invariants associated to $M$ and show that they determine it up to order 5. We then study the flat geometry (contact with planes) of a generic cuspidal cross-cap by classifying submersions which preserve it and relate the singularities of the resulting height functions with the geometric invariants.

We study geometric properties of caustics of pseudo-spherical surfaces, that is, surfaces with constant negative Gaussian curvature -1 in the Euclidean 3-space ℝ 3 {\mathbb{R}^{3}} . We investigate the Gaussian and the mean curvature of caustics of pseudo-spherical surfaces. Moreover, a certain condition required for the caustics to be minimal surfaces is derived.

A method to derive conservation laws for evolution equations that describe pseudospherical surfaces is introduced based on a geometrical property of these surfaces. A new third-order evolution equation is obtained as a first example for a nongeneric case in the classification given by Chern and Tenenblat [Stud. Appl. Math. 74, 1 (1986)].

In this paper, a lens distortion correction method for low-cost digital camera is proposed. The distortion coefficient and distortion center are estimated by using geometric invariants of perspective projection. The geometric invariants, including cross ratio of collinear points and straight-parallel-perpendicular lines, will be invariant in transforming from the world coordinate system to image coordinate system if there exists no distortion. We derive new distortion measure that is based on these geometric properties and can be optimized with nonlinear search technique. The method is easy to apply and lead to robust results with moderate effort. We verify accuracy and efficiency from experiments.KeywordsLens distortion correctionradial distortiongeometric invariantscross ratio

The main work of this paper is to investigate two kinds of generalized focal surfaces and two kinds of evolutes generated by spacelike curve lying in lightlike surfaces in Minkowski three‐space. Applying the method of unfolding theory in singularity theory to our study, it is shown that there exist the cuspidal edge and the swallowtail types of singularities in each of two classes of generalized focal surfaces under certain conditions; the only cusps will appear in each of evolutes. Two new geometric invariants are presented to classify the singularities of generalized focal surfaces and evolutes. Much more importantly, we reveal the correspondence among the geometric invariants, the types of singularities on generalized focal surfaces and evolutes, the singularities of two kinds of evolutes, and the contact of with the osculating spheres. Finally, several examples are presented to demonstrate the correctness of the theoretical results.

We show a relation between sign of Gaussian curvature of a cuspidal edge and geometric invariants via singularities of Gauss maps. Moreover, we define and characterize positivity/negativity of cusps of Gauss maps by geometric invariants, and show a relation between the signs of cusps and the Gaussian curvature.

The presence of noncyclic geometric invariant is revealed in all the phenomena where particle generation from vacuum or vacuum condensates appear. Aharonov-Anandan invariants then can help to study such systems and can represent a new tool to be used in order to provide laboratory evidence of phenomena particulary hard to be detected, such as Hawking and Unruh effects and some features of quantum field theory in curved space simulated by some graphene morphologies. It is finally suggested that a very precise quantum thermometer can be built by exploiting geometric invariants properties.

Exact nonadiabatic quantum evolution preserves many geometric properties of the molecular Hilbert space. In the first paper of this series ["Paper I," S. Choi and J. Vaníček, J. Chem. Phys. 150, 204112 (2019)], we presented numerical integrators of arbitrary-order of accuracy that preserve these geometric properties exactly even in the adiabatic representation, in which the molecular Hamiltonian is not separable into kinetic and potential terms. Here, we focus on the separable Hamiltonian in diabatic representation, where the split-operator algorithm provides a popular alternative because it is explicit and easy to implement, while preserving most geometric invariants. Whereas the standard version has only second-order accuracy, we implemented, in an automated fashion, its recursive symmetric compositions, using the same schemes as in Paper I, and obtained integrators of arbitrary even order that still preserve the geometric properties exactly. Because the automatically generated splitting coefficients are redundant, we reduce the computational cost by pruning these coefficients and lower memory requirements by identifying unique coefficients. The order of convergence and preservation of geometric properties are justified analytically and confirmed numerically on a one-dimensional two-surface model of NaI and a three-dimensional three-surface model of pyrazine. As for efficiency, we find that to reach a convergence error of 10-10, a 600-fold speedup in the case of NaI and a 900-fold speedup in the case of pyrazine are obtained with the higher-order compositions instead of the second-order split-operator algorithm. The pyrazine results suggest that the efficiency gain survives in higher dimensions.

We investigate the local differential geometric invariants of cuspidal edge and swallowtail from the view point of singularity theory. We introduce finite type invariants of such singularities (see Remark 1.5 and Theorem 2.11) based on certain normal forms for cuspidal edge and swallowtail. Then we discuss several geometric aspects based on our normal form. We also present several asymptotic formulas concerning our invariants with respect to Gauss curvature and mean curvature.