# leto/thesis

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 \section{Introduction} The propagation of longitudinal deformation waves in elastic rods is governed (\cite{LCZ}, \cite{Runz}, \cite{WM}) by the Generalized Pochammer-Chree Equations: \begin{equation}\label{eq:GPC1} \left( u - u_{xx} \right)_{tt} - \left( a_1 u + a_2 u^2 + a_3 u^3 \right)_{xx} =0 \end{equation} and \begin{equation} \label{eq:GPC2} \left( u - u_{xx} \right)_{tt} - \left( a_1 u + a_3 u^3 + a_5 u^5 \right)_{xx} =0 \end{equation} corresponding to different constitutive relations. References \cite{LCZ}, \cite{Runz}, \cite{WM} also discuss the primary references, including derivations and applications of these equations in various fields. In addition, motivated by experimental and numerical results, there are derivations of special families of solitary wave solutions by the extended $Tanh$ method \cite{LCZ}, and other ansatzen \cite{WM}. These extend earlier solitary wave solutions given by Bogolubsky \cite{Bogo} and Clarkson et. al \cite{CLVS} for special cases of \eqref{eq:GPC1} and \eqref{eq:GPC2}. In addition, \cite{Runz} generalizes the existence results in \cite{Sax} for solitary waves of \eqref{eq:GPC1} and \eqref{eq:GPC2}. In this paper, we initiate a fresh approach to the solitary wave solutions of the Generalized Pochammer-Chree equations \eqref{eq:GPC1} and \eqref{eq:GPC2}. We invoke the theory of reversible systems and the method of normal forms to categorize the possible solitary waves of \eqref{eq:GPC1} and \eqref{eq:GPC2} much more completely than done so far. As we shall see, several families of solitary waves exist in various regions of parameter space. Our main focus here will be on delineating the possible occurrence and multiplicity of solitary waves in different parameter regimes. Certain delicate questions relating to specific waves or wave families will form the basis of future work. The remainder of this paper is organized as follows. In Section 2, we delineate the possible families of solitary waves in various parameter domains and on certain important curves using the theory of reversible systems. In Sections 3 and 4, we next focus on the various transition curves and derive normal forms in their vicinity to confirm the existence of families of regular or delocalized solitary wave solutions in their vicinity.