Sobre uma família de equações de evolução não lineares : existência, classificação e instabilidade de soluções ondas viajantes

Abstract

This thesis is concerned with the orbital instability for a specific class of periodic traveling wave solutions with the mean zero related to the modified Camassa-Holm equation. These solutions, called snoidal waves, are written in terms of the Jacobi elliptic function sn. To prove these results we use the abstract methods of Grillakis, Shatah and Strauss, and the Floquet theory for periodic eigenvalue problems. Moreover, we classify all traveling wave solutions of the modified Camassa-Holm equation in the weak sense via parametrization of their maxima, minima and wave velocity constants, using the qualitative method of Lenells. This equation is shown to admit in addition to more popular solutions like smooth traveling waves and peakons, some not so well-known traveling waves as, for example, kinks, cuspons, composite waves and stumpons.