Variational techniques for breathers in nonlinear wave equations

In this thesis we investigate the quasilinear wave equation
\begin{align*}
g(x)w_{tt}-w_{xx}+h(x)(w_t^3)_t=0 \quad \text{for } (x,t)\in\mathbb{R}\times\mathbb{R},
\end{align*}
and the semilinear wave equation
\begin{align}
V(x)u_{tt}-\Delta u=f(x,t,u),\qquad\text{on}~~(x,t)\in\mathbb{R}^N\times\mathbb{R},
\end{align}
where $\Delta$ denotes the Laplacian acting only on the variable $x$. Most of the time we refer to $x$ as $\textit{space}$ and to $t$ as $\textit{time}$. We are specially interested in spatially localized and time-periodic solutions, so-called $\textit{breathers}$.