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\section{Normal form near $C_0$: solitary wave solutions}
Using \eqref{eq:linode}, the curve $C_0$, corresponding to $\lambda = 0,0,\pm \tilde{ \lambda } $, is given by
\begin{equation}
C_0: { p=0, q > 0 }
\end{equation}
Using \eqref{eq:qdef} implies
\begin{equation}
a_1 < c^2
\end{equation}
Denoting $\phi$ by $y_1$, \eqref{eq:ode} may be written as the two systems
\begin{subequations}\label{eq:system}
\begin{eqnarray}
\frac{d y_1 }{d z} &=& y_2 \\
\frac{d y_2 }{d z} &=& y_3 \\
\frac{d y_3 }{d z} &=& y_4 \\
\frac{d y_4 }{d z} &=& q y_3 - p y_1 - N_{1,2}(Y)
\end{eqnarray}
\end{subequations}
where
\begin{subequations}
\begin{eqnarray}
\mathcal{N}_1\left(Y\right) &=& - \frac{1}{c^2}\left[ 3 a_3 \left( 2 y_1 y_2^2 + y_1^2 y_3 \right) + 2 a_2\left( y_3 y_2 + y_2^2\right) \right] \\
\mathcal{N}_2\left(Y\right) &=& - \frac{1}{c^2}\left[ 3 a_3 \left( 2 y_1 y_2^2 + y_1^2 y_3\right) + 5 a_5 \left( 4 y_1^3 y_2^2 + y_1^4 y_3 \right) \right]
\end{eqnarray}
\end{subequations}
We wish to rewrite this as a first order reversible system in order to invoke the relevant theory \cite{IA}.
To that end, defining $Y=\left<y_1,y_2,y_3,y_4\right>^T$ equation \eqref{eq:system} may be written
\begin{equation}\label{eq:bilinear}
\frac{ dY }{ dz } = L_{pq} Y - G_{1,2}(Y,Y)
\end{equation}
where
\begin{equation}
L_{pq} = \left(
\begin{array}{cccc}
0&1&0&0\\
q/3&0&1&0\\
0&q/3&0&1\\
q^2 - p &0&q/3&0 \end{array} \right)
\end{equation}
Since $p=0$ for \eqref{eq:GPC1} and \eqref{eq:GPC2}, we have
\begin{equation} \label{eq:bilinear2}
\frac{ dY }{ dz } = L_{0q} Y - G_{1,2}(Y,Y)
\end{equation}
where
\begin{equation}\label{eq:nonlinear}
G_{1,2}(Y,Y) = \left<0,0,0,-\mathcal{N}_{1,2}\left(Y\right)\right>^T
\end{equation}
Next we calculate the normal form of \eqref{eq:bilinear2} near $C_0$. The procedure is
closely modeled on \cite{IA} and many intermediate steps may be found there.
\subsection{ Near $C_0$ }
Near $C_0$ the dynamics reduce to a two-dimensional Center Manifold
\begin{equation}\label{eq:c0cm}
Y = A \zeta_0 + B \zeta_1 + \Psi(\epsilon,A,B)
\end{equation}
and the corresponding normal form is
\begin{subequations}\label{eq:c0nf}
\begin{eqnarray}
\frac{dA}{dz} &=& B \label{eq:c0nfa} \\
\frac{dB}{dz} &=& b \epsilon A + \tilde{c} A^2 \label{eq:c0nfb}
\end{eqnarray}
\end{subequations}
Here,
\begin{equation}
\epsilon = \left( \frac{q^2}{9} - p\right) - \left(\frac{q}{3}\right)^2 = - p
\end{equation}
measures the perturbation around $C_0$, and
\begin{subequations}\label{eq:lineareigs}
\begin{eqnarray}
\zeta_0 &=& \left<1,0,-q/3,0\right>^T\\
\zeta_1 &=& \left<0,1,0,-2 q/3\right>^T
\end{eqnarray}
\end{subequations}
The linear eigenvalue of \eqref{eq:c0nf} satisfies
\begin{equation}\label{eq:lineig}
\lambda^2 = b \epsilon
\end{equation}
The characteristic equation of the linear part of
\eqref{eq:bilinear2} is
\begin{equation}\label{eq:charlinear}
\lambda^4 - q \lambda^2 - \epsilon = 0
\end{equation}
Hence, the eigenvalues near zero (the Center Manifold) satisfy $\lambda^4 \ll \lambda^2$ and hence
\begin{equation}\label{eq:lindominant}
\lambda^2 \sim -\frac{\epsilon}{q}
\end{equation}
Matching \eqref{eq:lineig} and \eqref{eq:lindominant} implies
\begin{equation}
b = - \frac{1}{q}
\end{equation}
and only the nonlinear coefficient $\tilde{c}$ remains to be determined in the normal form \eqref{eq:c0nf}.
In order to determine $\tilde{c}$ (the coefficient of $A^2$ in \eqref{eq:c0nf})
we calculate $\frac{dY}{dz}$ in two ways and match the $\mathcal{O}(A^2)$
terms. To this end, using the standard 'suspension' trick of treating the
perturbation parameter $\epsilon$ as a variable, we expand the function $\Psi$
in \eqref{eq:c0cm} as
\begin{equation}\label{eq:psiexp}
\Psi(\epsilon,A,B) = \epsilon A \Psi_{10}^1 + \epsilon B \Psi_{01}^1 + A^2 \Psi_{20}^0 + A B \Psi_{11}^0 + B^2 \Psi_{02}^0 + \cdots
\end{equation}
where the subscripts denote powers of $A$ and $B$, respectively, and the
superscript denotes the power of $\epsilon$. In the first way of computing
$dY/dz$, we take the $z$ derivative of \eqref{eq:c0cm} (using \eqref{eq:c0nf}
and \eqref{eq:psiexp}). The coefficient of $A^2$ in the resulting expression
is $\tilde{c} \zeta_1 $. In the second way of computing $dY/dz$, we use
\eqref{eq:c0cm} and \eqref{eq:psiexp} in \eqref{eq:bilinear}. The coefficient
of $A^2$ in the resulting expression is $ L_{0,q} \Psi_{20}^0 -
G_{1,2}\left(\zeta_0,\zeta_0\right)$. Hence
\begin{equation}\label{eq:A2coef}
\tilde{c} \zeta_1 = L_{0q} \Psi_{20}^0 - G_{1,2}(\zeta_0,\zeta_0) \end{equation}
Using \eqref{eq:lineareigs} and \eqref{eq:nonlinear} and denoting
$\Psi_{20}^0 = \left<x_1,x_2,x_3,x_4\right>$ in \eqref{eq:A2coef} yields the equations
\begin{subequations}
\begin{eqnarray}
0 &=& x_2 \\
\tilde{c} &=& \frac{q}{3} x_1 + x_3 \label{eq:A2coefb}\\
0 &=& \frac{q}{3} x_2 + x_4 \implies x_4 = 0
\textrm{ using \eqref{eq:A2coefb} }
\end{eqnarray}
\end{subequations}
and
\begin{equation}
-\frac{2q}{3} \tilde{c} = \frac{q}{3}\left(\frac{q}{3} x_1 + x_3 \right) + \frac{q}{3 c^2}\left( 3 a_3 + 5 a_5 \right) = \frac{q}{3}\tilde{c} + \frac{q}{3 c^2}\left( 3 a_3 + 5 a_5 \right)
\textrm{ using \eqref{eq:A2coefb} }
\end{equation}
Hence we obtain
\begin{equation}
\tilde{c} = - \frac{1}{3 c^2} \left(3 a_3 + 5 a_5 \right)
\end{equation}
Therefore, the normal form near $C_0$ is
\begin{subequations}\label{eq:c0nfgpc1}
\begin{eqnarray}
\frac{dA}{dz} &=& B \\
\frac{dB}{dz} &=& -\frac{\epsilon}{q} A - \frac{a_3}{ c^2} A^2
\end{eqnarray}
\end{subequations}
for \eqref{eq:GPC1} and
\begin{subequations}\label{eq:c0nfgpc2}
\begin{eqnarray}
\frac{dA}{dz} &=& B \\
\frac{dB}{dz} &=& -\frac{\epsilon}{q} A - \frac{1}{3 c^2} \left(3 a_3 + 5 a_5 \right) A^2
\end{eqnarray}
\end{subequations}
for \eqref{eq:GPC2}.
The normal form \eqref{eq:c0nfgpc1} admits a homoclinic solution (near $C_0$) of the form
\begin{equation} \label{eq:soliton1}
A\left(z\right) = \ell \space \mathrm{sech}^2\left(k z\right)
\end{equation}
with
\begin{subequations}
\begin{eqnarray}
k &=& \sqrt{\frac{-\epsilon}{4q}} \\
\ell &=& \frac{ - 3 \epsilon c^2 }{2 q a_3 }
\end{eqnarray}
\end{subequations}
Similarly, the normal form \eqref{eq:c0nfgpc2} admits a homoclinic solution (near $C_0$) of the form
\begin{equation}\label{eq:soliton2}
A\left(z\right) = \ell \space \mathrm{sech}^2\left(k z\right)
\end{equation}
with
\begin{subequations}
\begin{eqnarray}
k &=& \sqrt{\frac{-\epsilon}{4q}} \label{eq:keq} \\
\ell &=& \frac{ - 9 \epsilon c^2 }{2 q \left(3 a_3 + 5 a_5\right) }
\end{eqnarray}
\end{subequations}
Hence, since $\epsilon = - p $, and the curve $C_0$ corresponds to $p=0,q>0$, solitary waves of the
form \eqref{eq:soliton1} exist in the vicinity of $C_0$ for
\begin{equation}
p > 0, q > 0
\end{equation}
which implies that $a_1 < c^2 $ (such that $k$ in \eqref{eq:keq} is real.) As mentioned in section 2, one may show the persistence
of this homoclinic solution in the original traveling wave ODE \eqref{eq:linode}. Thus, we have
demonstrated the existence of solitary waves of \eqref{eq:GPC2} for $p=0^+, q>0$.
Similarly, the curve $C_1$ corresponds to $p=0,q<0$, solitary waves of the form \eqref{eq:soliton1} exist
in the vicinity of $C_1$ for
\begin{equation}
p < 0, q < 0
\end{equation}
which implies $ a_1 > c^2 $.
Again, one may show the persistence
of this homoclinic solution in the original traveling wave ODE \eqref{eq:linode}. Thus, we have
demonstrated the existence of solitary waves of \eqref{eq:GPC2} for $p=0^-, q<0$.