Estabilidade de ondas periódicas para modelos dispersivos não-lineares

Abstract

This thesis is concerned with the qualitative study of the orbital and linear stability associated with three non-linear models of evolution equations. Initially, by using the abstract theory due to Grillakis, Shatah and Strauss, [37], we determine the orbital instability of periodic dnoidal waves at the Sobolev space constituted by even periodic functions. Next, by using the classical variational theory as in Benjamin [14], Bona [17] and Weinstein [66] (see also Angulo [8]), we show the orbital stability of periodic waves in 1 per ([0, L]) related to the Logarithmic-Schrödinger equation. This study improves previous results proposed by Natali and Neves, [55]. These authors have obtained the orbital stability of periodic waves by restricting the approach to the even periodic Sobolev space. In both models, we present the spectral analysis of the associated Hill?s operators, by using an adaptation of Floquet?s theory due to Natali and Neves (see [55]) and Natali and Pastor (see [56]). Finally, we study the linear stability of periodic traveling waves to ILW Equation (Intermediate Long Wave Equation). We consider explicit periodic waves with the mean zero property in order to use the Hamiltonian-Krein index?s theory given by Deconinck and Kapitula in [31]