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Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides, with Metamaterials, Having Parabolic Law Non-Linearity
Keywords:Integrable system, exact solution, homoclinic and heteroclinic orbit, periodic solution, peakon, compacton.
      Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity is investigated by the method of dynamical systems. Because the functions $\phi(\xi)$ in the solutions $q(x,t)=\phi(\xi)\exp(i(-kx+\omega t)),\ (\xi=x-vt)$ satisfy a singular planar dynamical system having two singular straight lines. By using the bifurcation theory method of dynamical systems to the equations of $\phi(\xi)$, under 28 different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons and peakons as well as compacton solutions for this planar dynamical system are obtained. 62 exact explicit solutions of system (5) are derived.