# public leto /thesis

### Subversion checkout URL

You can clone with HTTPS or Subversion.

Fetching contributors…

Cannot retrieve contributors at this time

executable file 53 lines (43 sloc) 2.602 kb
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 \chapter{CHAPTER FOUR: RESULTS} \label{chapter_4}In this thesis, we apply a recently developed technique tocomprehensively categorize all possible families of solitary wavesolutions in two models of topical interest.The models considered are:\begin{itemize}\item the Generalized Pochhammer-Chree Equations, which govern the propagation of longitudinal waves in elastic rods,and\item a generalized microstructure PDE.\end{itemize}Limited analytic results exist for the occurrence of one family of solitarywave solutions for the Microstructure equation and results using a Hamiltonianformulation have recently been found in the Pochhammer-Chree equations \cite{LiZhang}. Since, asmentioned above, solitary wave solutions often play a central role in thelong-time evolution of an initial disturbance, we consider such solutions ofboth models here (via the normal form approach) within the framework ofreversible systems theory.Besides confirming the existence of the known family of solitary waves for eachmodel, of the form\begin{equation} A\left(z\right) = \ell \space \mathrm{sech}^2\left(k z\right)\end{equation}we find a continuum of delocalized solitary waves (or homoclinics tosmall-amplitude periodic orbits). On isolated curves in the relevant parameterregion, the delocalized waves reduce to genuine embedded solitons. These solitary waves are called delocalized because they have exponentiallysmall oscillations as $|z|\rightarrow\infty$ and so are not localized in space.This is often referred to as a soliton in a "sea of radiation."These curves are defined by the behavior of the four eigenvalues of the characteristicequation $\lambda^4 - q \lambda^2 - \epsilon = 0$. Specifically, the multiplicity of the eigenvalues change as the parameters vary across these curves.Thus, these curves define separatrices between vastly different dynamics in the traveling wave ODE as well as the original PDE.One may easily verify that $\lim_{z\rightarrow\pm\infty} A(z) = 0$, therefore$A(z)$ compromises a homoclinic orbit, since it connects the fixed point $0$ toitself. The importance of homoclinic orbits in the traveling wave ODE is thatthey correspond to soliton pulse solutions of the original PDE \cite{IA}. Iooss\& P\'erou\eme have proved that solutions in the traveling wave ODE persist inthe original system \cite{IP} for reversible 1:1 resonance vector fields. Forthe microstructure equation, the new family of solutions occur in regions ofparameter space distinct from the known solitary wave solutions and are thusentirely new. %42`
Something went wrong with that request. Please try again.