# leto/thesis

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 \section{Solitary waves: local bifurcation} Solitary waves of \eqref{eq:GPC1} and \eqref{eq:GPC2} of the form $v(x,t) = \phi\left(x - c t\right) = \phi\left(z\right)$ satisfy the fourth-order traveling wave ODE \begin{equation} \label{eq:ode} \phi_{zzzz} - q \phi_{zz} + p \phi = \mathcal{N}_{1,2}[\phi] \end{equation} where \begin{subequations} \begin{eqnarray} \mathcal{N}_1\left[\phi\right] &=& - \frac{1}{c^2}\left[ 3 a_3 \left( 2 \phi \phi_z^2 + \phi^2 \phi_{zz} \right) + 2 a_2\left( \phi_{zz} \phi_z + \phi_z^2\right) \right] \\ \mathcal{N}_2\left[\phi\right] &=& - \frac{1}{c^2}\left[ 3 a_3 \left( 2 \phi \phi_z^2 + \phi^2 \phi_{zz}\right) + 5 a_5 \left( 4 \phi^3 \phi_z^2 + \phi^4 \phi_{zz} \right) \right] \end{eqnarray} \end{subequations} \begin{subequations} \begin{eqnarray} z &\equiv& x - c t\\ p &\equiv& 0\label{eq:pdef} \\ q &\equiv & 1 - \frac{a_1}{c^2} \label{eq:qdef} \\ \end{eqnarray} \end{subequations} Equation \eqref{eq:ode} is invariant under the transformation $z \mapsto -z$ and is thus a reversible system. In this section we shall use the theory of reversible systems to characterize the homoclinic orbits to the fixed point of \eqref{eq:ode}, which correspond to pulses or solitary waves of \eqref{eq:GPC1} and \eqref{eq:GPC2} in various regions of the $(p,q)$ plane. The linearized system corresponding to \eqref{eq:ode} \begin{equation} \label{eq:linode} \phi_{zzzz} - q \phi_{zz} + p \phi = 0 \end{equation} has a fixed point \begin{equation}\label{eq:fp} \phi = \phi_z = \phi_{zz} = \phi_{zzz} = 0 \end{equation} Solutions $\phi = k e^{\lambda x}$ satisfy the characteristic equation $\lambda^4 - q \lambda^2 + p = 0$ from which one may deduce that the structure of the eigenvalues is distinct in two regions of $\left(p,q\right)$-space. Since $p=0$ we have only two possible regions of eigenvalues. We denote $C_0$ as the positive $q$ axis and $C_1$ the negative $q$-axis. First we shall consider the bounding curves $C_0$ and $C_1$ and their neighborhoods, then we shall discuss the possible occurrence and multiplicity of homoclinic orbits to \eqref{eq:fp}, corresponding to pulse solitary waves of \eqref{eq:GPC1} and \eqref{eq:GPC2}, in each region: \begin{description} \item[Near $C_0$] The eigenvalues have the structure $\lambda_{1-4} = 0,0,\pm \lambda$, ($\lambda \in \mathbb{R}$) and the fixed point \eqref{eq:fp} is a saddle-focus. \item[Near $C_1$] Here the eigenvalues have the structure $\lambda_{1-4} = 0,0,\pm i \omega$, ($\omega \in \mathbb{R}$) . We will show by analysis of a four-dimensional normal form in Section 4 that there exists a $\mathrm{sech}^2$ homoclinic orbit near $C_1$. \end{description} Having outlined the possible families of orbits homoclinic to the fixed point \eqref{eq:fp} of \eqref{eq:linode}, corresponding to pulse solitary waves of \eqref{eq:GPC1} and \eqref{eq:GPC2}, we now derive normal forms near the transition curves $C_0$ and $C_1$ to confirm the existence of regular or delocalized solitary waves in the corresponding regions of $\left(p,q\right)$ parameter space.