leto/thesis

Switch branches/tags
Nothing to show
Fetching contributors…
Cannot retrieve contributors at this time
217 lines (181 sloc) 7.33 KB
 \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 $Y=\left^T$ 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 $\Psi_{20}^0 = \left$ 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 $p=0^-, q<0$.