A rigorous derivation of the asymptotic wavenumber of spiral wave solutions of the complex Ginzburg-Landau equation

In this work n-armed Archimedian spiral wave solutions of the complex Ginzburg-Landau equation are considered. These solutions are showed to depend on two characteristic parameters, the so called twist parameter and the asymptotic wavenumber. The existence and uniqueness of the value of the asymptotic wavenumber, depending on the twist parameter, for which n-armed Archimedian spiral wave solutions exist is a classical result, obtained back in the 80s by Kopell and Howard. In this work we deal with a different problem, that is, the asymptotic expression of the asymtptotic wavenumer for small values of the twist parameter. Since the eighties, different heuristic perturbation techniques, like formal asymptotic expansions, have conjectured an asymptotic expression of which is exponentially small with respect to the twist parameter. However, the validity of this expression has remained opened until now, despite of the fact that it has been widely used for more than 40 years. In this work, using a functional analysis approach, we finally prove the validity of the asymptotic formula, providing a rigorous bound for its relative error

Dynamics of spiral waves in the complex Ginzburg-Landau equation in bounded domains

Multiple-spiral-wave solutions of the general cubic complex Ginzburg-Landau equation in bounded domains are considered. We investigate the effect of the boundaries on spiral motion under homogeneous Neumann boundary conditions, for small values of the twist parameter $q$. We derive explicit laws of motion for rectangular domains and we show that the motion of spirals becomes exponentially slow when the twist parameter exceeds a critical value depending on the size of the domain. The oscillation frequency of multiple-spiral patterns is also analytically obtained

Motion of spiral waves in the Complex Ginzburg-Landau equation

Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered. For parameters close to the vortex limit, and for a system of spiral waves with well-separated centres, laws of motion of the centres are found which vary depending on the order of magnitude of the separation of the centres. In particular, the direction of the interaction changes from along the line of centres to perpendicular to the line of centres as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wavenumber and frequency are determined. These depend on the positions of the centres of the spirals, and so evolve slowly as the spirals move

Bullying and gender violence at school: analysis of teacher perception

Nowadays gender violence and bullying continue to be considered a matter of great concern to society. This study analyses the teachers'' perceptions of these two social problems in the educational context. The interventions that are carried out from the school to prevent and respond to these behaviours are studied. From a methodological point of view, the study responds to the paradigm of qualitative research. We interviewed fifteen teachers of Early Childhood Education, Primary Education and Secondary Education of the Spanish educational system. The information collected indicates that teachers advocate a necessary reinforcement of emotional competencies and social skills from schools. In addition, they express their dissatisfaction due to the insufficient strategies applied from the school to respond to these violent behaviours. These results support any initiative that aims to train teachers to prevent possible cases of bullying or gender violence

Monografia: Educació i Telemàtica

La referencia a les telecomunicacions en l'àmbit educatiu ha augmentat notablement a causa, sobretot, de les expectatives que l'ús d'lnternet ha despertat entre professionals, administradors, etc. El fenòmen és recent, per aquest motiu el tipus de publicació se centra bé en la descripció de les possibilitats bé en l'analisi d'experiencies concretes. La composició del número monografic Educació i Telematica es mou en aquesta línia. En aquest sentit, hem intentat recollir articles de reflexió juntament amb experiencies concretes que s'estan portant a terme amb l'objectiu que el lector tingui una visió general de les possibilitats pedagógiques de I'ús de les xarxes de comunicació en l'àmbit educatiu

Motion of spiral waves in the complex Ginzburg-Landau equation

Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered. For parameters close to the vortex limit, and for a system of spiral waves with well-separated centres, laws of motion of the centres are found which vary depending on the order of magnitude of the separation of the centres. In particular, the direction of the interaction changes from along the line of centres to perpendicular to the line of centres as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wavenumber and frequency are determined. These depend on the positions of the centres of the spirals, and so evolve slowly as the spirals move. © 2009 Elsevier B.V

Interaction of spiral waves in the complex Ginzburg-Landau equation.

Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered, and laws of motion for the centers are derived. The direction of the motion changes from along the line of centers to perpendicular to the line of centers as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wave number and frequency are also determined, which evolve slowly as the spirals move

Interaction of spiral waves in the complex Ginzburg-Landau equation.

Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered, and laws of motion for the centers are derived. The direction of the motion changes from along the line of centers to perpendicular to the line of centers as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wave number and frequency are also determined, which evolve slowly as the spirals move