# leto/thesis

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 b7c8ae0 initial checkin leto authored Dec 8, 2007 1 \section{Solitary waves: local bifurcation} b25d28c fixing wording in GPC sec 1/2 leto authored Dec 28, 2007 2 08fe4a0 abstract and sec.1 for GPC fixes from roy leto authored Jan 17, 2008 3 Solitary waves of \eqref{eq:GPC1} and \eqref{eq:GPC2} of the form b25d28c fixing wording in GPC sec 1/2 leto authored Dec 28, 2007 4 $v(x,t) = \phi\left(x - c t\right) = \phi\left(z\right)$ 0dd662c lots of spelling corrections thanks to ispell -t leto authored Feb 8, 2008 5 satisfy the fourth-order traveling wave ODE b25d28c fixing wording in GPC sec 1/2 leto authored Dec 28, 2007 6 \begin{equation} \label{eq:ode} \phi_{zzzz} - q \phi_{zz} + p \phi = \mathcal{N}_{1,2}[\phi] 7 \end{equation} 8 9 where 10 \begin{subequations} 11 \begin{eqnarray} 12 \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] \\ 13 \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] 14 \end{eqnarray} 15 \end{subequations} 16 17 \begin{subequations} 18 \begin{eqnarray} 19 z &\equiv& x - c t\\ 20 p &\equiv& 0\label{eq:pdef} \\ 21 q &\equiv & 1 - \frac{a_1}{c^2} \label{eq:qdef} \\ 22 \end{eqnarray} 23 \end{subequations} 24 25 Equation \eqref{eq:ode} is invariant under the transformation $z \mapsto -z$ and is thus a reversible system. In this section we shall 26 use the theory of reversible systems to characterize the homoclinic orbits to the fixed point of \eqref{eq:ode}, which correspond to pulses 3e22f86 finalish corrections after printout and intense eye-popping scrutiny leto authored Feb 12, 2008 27 or solitary waves of \eqref{eq:GPC1} and \eqref{eq:GPC2} in various regions of the $(p,q)$ plane. b25d28c fixing wording in GPC sec 1/2 leto authored Dec 28, 2007 28 29 The linearized system corresponding to \eqref{eq:ode} 30 \begin{equation} 31 \label{eq:linode} \phi_{zzzz} - q \phi_{zz} + p \phi = 0 32 \end{equation} 33 has a fixed point \begin{equation}\label{eq:fp} \phi = \phi_z = \phi_{zz} = \phi_{zzz} = 0 \end{equation} 34 35 Solutions $\phi = k e^{\lambda x}$ satisfy the characteristic equation 36 $\lambda^4 - q \lambda^2 + p = 0$ from which one may deduce that the structure 37 of the eigenvalues is distinct in two regions of $\left(p,q\right)$-space. 38 Since $p=0$ we have only two possible regions of eigenvalues. We denote $C_0$ 39 as the positive $q$ axis and $C_1$ the negative $q$-axis. First we shall 0dd662c lots of spelling corrections thanks to ispell -t leto authored Feb 8, 2008 40 consider the bounding curves $C_0$ and $C_1$ and their neighborhoods, then we shall discuss the possible 41 occurrence and multiplicity of homoclinic orbits to \eqref{eq:fp}, corresponding 095afd8 lots of fixes from roy and lookig over a printed version. almost done… leto authored Jan 18, 2008 42 to pulse solitary waves of \eqref{eq:GPC1} and \eqref{eq:GPC2}, in each region: b25d28c fixing wording in GPC sec 1/2 leto authored Dec 28, 2007 43 44 \begin{description} 45 \item[Near $C_0$] 46 The eigenvalues have the structure $\lambda_{1-4} = 0,0,\pm \lambda$, ($\lambda \in \mathbb{R}$) and the fixed point 47 \eqref{eq:fp} is a saddle-focus. 48 \item[Near $C_1$] 3e22f86 finalish corrections after printout and intense eye-popping scrutiny leto authored Feb 12, 2008 49 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 b1b1ff1 change to article.cls and put contact info leto authored Feb 13, 2008 50 four-dimensional normal form in Section 4 that there exists a $\mathrm{sech}^2$ homoclinic orbit near $C_1$. b25d28c fixing wording in GPC sec 1/2 leto authored Dec 28, 2007 51 \end{description} 52 53 Having outlined the possible families of orbits homoclinic to the fixed point \eqref{eq:fp} of \eqref{eq:linode}, 095afd8 lots of fixes from roy and lookig over a printed version. almost done… leto authored Jan 18, 2008 54 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 $C_1$ b25d28c fixing wording in GPC sec 1/2 leto authored Dec 28, 2007 55 to confirm the existence of regular or delocalized solitary waves in the corresponding regions of $\left(p,q\right)$ parameter space. 56