Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Non-Linearity
Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems
Traveling wave solutions of a modified vector-disease model
Title: Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides, with Metamaterials,Having Polynomial Law Non-Linearity
Abstract: Raman soliton model in nanoscale optical waveguides, with metamaterials, having polynomial law non-linearity are 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 23 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 be given. Under different parameter conditions, solutions $q(x,t)$ can be exactly obtained. 92 exact explicit solutions of system (6) are derived
Title: Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems
Abstract: This paper deals with the problem of periodic orbit bifurcations for high-dimensional piecewise smooth systems. Under the assumption that the unperturbed system has a family of periodic orbits which are transversal to the switch plane, a formula for the first order Melnikov vector function is developed which can be used to study the number of periodic orbits bifurcated from the periodic orbits. We especially can use the function to study the number of periodic orbits both in degenerate Hopf bifurcations and in degenerate homoclinic bifurcations. Finally, we present two examples to illustrate an application of the theoretical results.
Title: Traveling wave solutions of a modified vector-disease model
Abstract: We discuss the existence and asymptotic behavior of traveling wave fronts in a modified vector-disease model. We first establish the existence of traveling wave solutions for the modified vector-disease model without delay, then the existence of traveling fronts for the model with a special local delay convolution kernel are obtained by employing geometric singular perturbation theory and the linear chain trick. At last, we investigate the local stability of the steady states, the existence and the asymptotic behavior of traveling wave solutions for that model with a special non-local delay convolution kernel.