On some Nonlinear Aspects of Wave Motion

Abstract: In the first part of this thesis we consider the governing equations for capillary water waves given by the Euler equations with a free surface under the influence of surface tension over a flat bottom. We look for two-dimensional steady periodic waves. The problem is first transformed to a nonlinear elliptic equation in a rectangle. Using bifurcation and degree theory we then prove the existence of a global continuum of such waves. In the second part of the thesis we inverstigate an equation which is a model for shallow water waves and waves in a circular cylindrical rod of a compressible hyperelastic material. We present sufficient conditions for global existence and blow-up.