Abstract

Resumen El procedimiento para resolver ecuaciones diferenciales ordinarias mediante los llamados grupos de Lie se basa en la geometr´ıa diferencial y su alcance notable, una vez que se dominan las t´ecnicas elementales en esta disciplina. La resoluci´on de algunos ejemplos nos ayudar´an en este cometido. 

Abstract

El cálculo de variaciones es una herramienta matem´atica muy importante que ,en algunos casos,se emplea para resolver problemas en la naturaleza de contenido práctico e incluso teórico.Es por ello que presentamos algunos ejemplos sencillos resueltos completamente con la llamada teoría de Jacobi de los puntos conjugados

Abstract

The nonlinear and unsteady dynamics of the edge waves is discussed. Two physical processes: dispersive focusing and nonlinear modulational instability, are studied. Both mechanisms can induce the appearance of the short-living large-amplitude isolated waves and intense wave packets (“freak edge waves”).

Abstract

This work os a kind of homage to Sophus Lie, using a sugestion from Isaac Newton that wrote that was beter to give examples than precepts. So, I study contact transformations through the well-known parabolic trajectories.

Abstract

The nonlinear and unsteady dynamics of the edge waves is discussed. Two physical processes: dispersive focusing and nonlinear modulational instability, are studied. Both mechanisms can induce the appearance of the short-living large-amplitude isolated waves and intense wave packets (“freak edge waves”).

Abstract

This paper deals with the possible involvement of freak, rogue or giant waves in the damage suffered by the Tanker Prestige, which eventually led to its sinking. By reason of their very characteristics, giant waves are hard to record. Their more or less sudden appearance makes them fairly elusive objects, except for the consequences they produce. However, some hints with regard to the probability of their occurrence can be derived from considering how close the maritime weather of the area of interest is to the situation which is optimal for their generation. This paper takes into account the wave field in the area and at the time of the Prestige accident and investigates how near or how far the wave field was to the instability conditions that are favourable to the generation of freak waves in the different approximations. This paper explores mostly the modulation instability which is one of the most common mechanisms to produce freak waves: it leads to the decomposition of an initially homogeneous train of Stokes waves firstly into a series of groups of waves, whose envelope then produces the so-called “solitons”, that then collapse in the form of a giant wave. This mechanism mainly occurs in deep waters. Zakharov studied it in 1968 and, independently, Benjamin and Feir analysed it in 1967. This paper proves that the wave field was conspicuously two dimensional with two main wave components travelling in directions almost orthogonal to each other. This means that the wave field was well outside the instability domain. Therefore it is concluded that freak waves were very unlikely generated and it is improbable that they were responsible for the accident. Correspondence to: A. Lechuga (antonio.lechuga@cedex.es)1Introduction Some people and institutions concerned with the accident of the Tanker Prestige expressed the opinion that perhaps the main damage that eventually led to its sinking could be caused by freak waves. In order to clarify this point we are going to study the possible involvement of such waves in the damage suffered by the Prestige, using observed wave records and theoretical models to describe the sea conditions. Firstly, it is useful to define what we understand as a freak wave. To this end we will take into account the descriptions appearing in the literature and the opinions shared by many researchers, such as Pelinovsky and Kharif (2000), Peregrine (1979), Osborne et al. (2000), Zakharov et al. (2006), and Dyachencko and Zakharov (2005), amongst many others. The two last named authors, in particular, point out the following characteristics of freak waves:

Abstract

This work is a kind of homage to Sophus Lie using a sugestion given by Isaac Newton: To show theory through examples instead of precepts. So contact transformations are studying by the well-known parabolic trajectories.

Abstract

This work is a kind of homage to the great mathematicien Sophus Lie , but following the sugestion of Isaac Newton I use an example Instead of theory. The possibilities of infinitesimal groups in the solución of dlfferential equations is well-known todas.