Angular Pseudomomentum Theory for the Generalized Nonlinear Schr\"{o}dinger Equation in Discrete Rotational Symmetry Media

We develop a complete mathematical theory for the symmetrical solutions of the generalized nonlinear Schr\"odinger equation based on the new concept of angular pseudomomentum. We consider the symmetric solitons of a generalized nonlinear Schr\"odinger equation with a nonlinearity depending on the modulus of the field. We provide a rigorous proof of a set of mathematical results justifying that these solitons can be classified according to the irreducible representations of a discrete group. Then we extend this theory to non-stationary solutions and study the relationship between angular momentum and pseudomomentum. We illustrate these theoretical results with numerical examples. Finally, we explore the possibilities of the generalization of the previous framework to the quantum limit.Comment: 18 pages; submitted to Physica

A topological charge selection rule for phase singularities

We present an study of the dynamics and decay pattern of phase singularities due to the action of a system with a discrete rotational symmetry of finite order. A topological charge conservation rule is identified. The role played by the underlying symmetry is emphasized. An effective model describing the short range dynamics of the vortex clusters has been designed. A method to engineer any desired configuration of clusters of phase singularities is proposed. Its flexibility to create and control clusters of vortices is discussed.Comment: 4 pages, 3 figure

Nonequilibrium quantum dynamics of partial symmetry breaking for ultracold bosons in an optical lattice ring trap

A vortex in a Bose-Einstein condensate on a ring undergoes quantum dynamics in response to a quantum quench in terms of partial symmetry breaking from a uniform lattice to a biperiodic one. Neither the current, a macroscopic measure, nor fidelity, a microscopic measure, exhibit critical behavior. Instead, the symmetry memory succeeds in identifying the point at which the system begins to forget its initial symmetry state. We further identify a symmetry energy difference in the low lying excited states which trends with the symmetry memory

Vorticity cutoff in nonlinear photonic crystals

Using group theory arguments, we demonstrate that, unlike in homogeneous media, no symmetric vortices of arbitrary order can be generated in two-dimensional (2D) nonlinear systems possessing a discrete-point symmetry. The only condition needed is that the non-linearity term exclusively depends on the modulus of the field. In the particular case of 2D periodic systems, such as nonlinear photonic crystals or Bose-Einstein condensates in periodic potentials, it is shown that the realization of discrete symmetry forbids the existence of symmetric vortex solutions with vorticity higher than two.Comment: 4 pages, 5 figures; minor changes in address and reference

Análisis competencial de una tarea de modelización abierta

Según diversos autores, las actividades dirigidas al desarrollo conjunto de las competencias matemáticas deben relacionarse con la realidad y los procesos de modelización matemática. El objetivo del presente trabajo es describir, a partir de la producción de un grupo de alumnos de tercer curso de ESO, las competencias que los alumnos han de poner en juego para resolver una tarea genuina y completa de modelización y que formarían parte de la propia competencia en Modelizació

Vortex transmutation

Using group theory arguments and numerical simulations, we demonstrate the possibility of changing the vorticity or topological charge of an individual vortex by means of the action of a system possessing a discrete rotational symmetry of finite order. We establish on theoretical grounds a "transmutation pass rule'' determining the conditions for this phenomenon to occur and numerically analize it in the context of two-dimensional optical lattices or, equivalently, in that of Bose-Einstein condensates in periodic potentials.Comment: 4 pages, 4 figure

On the construction of a geometric invariant measuring the deviation from Kerr data

This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed, to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation ---the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant ---however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.Comment: 36 pages. Updated references. Technical details correcte

Teacher s management of the modeling activity in the classroom at secondary level

[EN] the introduction of modeling in the secondary level demands the reconsideration about the teaching methodology that promotes and stimulates this activity in classroom. In this paper we present a classroom experience carried out in ninth grade. We analyze the work dynamic and the role of debate and interventions of the teacher in the process of solving the modeling tasks.[ES] La introducción de la modelización en secundaria requiere un replanteamiento de la metodología de enseñanza que promueva y posibilite la actividad modelizadora de nuestros alumnos. En el presente artículo, a través de una experiencia de aula llevada a cabo en tercero de secundaria, analizamos la dinámica de trabajo y el papel del debate y las intervenciones del profesor en el proceso de resolución de las tareas de modelización.Gallart-Palau, C.; Ferrando Palomares, I.; García-Raffi, LM. (2015). El profesor ante la actividad modelizadora en el aula de secundaria. Suma. (79):9-16. http://hdl.handle.net/10251/99867S9167

Large-Size Star-Shaped Conjugated (Fused) Triphthalocyaninehexaazatriphenylene

Star-shaped triphthalocyaninehexaazatriphenylene 1 was synthesized via condensation between a new building block 1,2-diaminophthalocyanine and cyclohexanehexaone. Compound 1 represents the largest star-shaped phthalocyanine-fused hexaazatriphenylene reported so far. This largely expanded phthalocyanine shows good solubility and has a strong tendency to aggregate in both solution and on surface, indicating its potential as active component in organic electronic devices.This research was financially supported by the Spanish Ministry of Economy and Competitiveness (Mineco) of Spain [CTQ2014-55798-R, CTQ2015-71936-REDT, CTQ2013-40480-R and “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV-2015-0496)], Generalitat Valenciana (Prometeo 2012/010), the Networking Research Center on Bioengineering, Biomaterials and Nanomedicine (CIBER-BBN) and by ERC StG 2012-306826 e-GAMES. A. C. acknowledges the Materials Science PhD program of UAB.Peer reviewe