| dc.description.abstract | In this thesis, we present an original existence proof of solitary wave solutions to a class of pseudodifferential evolution equations. We seek traveling waves with constant velocity $c$ of the form $u(x-ct)$ of
\begin{equation*}
u_t + (n(u) - Lu)_x = 0 \quad \text{in} \ \mathbb{R},
\end{equation*}
\ \\
through variational methods. By integrating over $\mathbb{R}$ with respect to the spatial variable, and assuming that the solution vanishes at infinity, we arrive at
\begin{equation*}
-cu + n(u) - Lu = 0 \quad \text{in} \ \mathbb{R},
\end{equation*}
which is the standing point for our analysis. We prove existence of solutions to these equations by the technique previously employed by Albert \cite{Albert} and Arnesen \cite{MA2} amongst others. Here, $n$ is a nonlinear term, and compared to what has earlier been studied, it is now inhomogeneous and includes a higher order term. $L$ is a Fourier multiplier operator of order $s \geq 0$. The higher order term included in the nonlinearity significantly changes the characteristics of the problem compared to what has previously been studied for this combination of equation and linear operator. We also introduce the principle of concentration compactness; the main ingredient in order to prove that we have compactness, despite working on an unbounded domain. | |
