Power-Logistic Shocks in a Lossy Nonlinear Transmission Line

Abstract

We consider a lossy transmission line with a nonlinear voltage--charge relation. In the framework of the inverse-design method, starting from a prescribed monotonic front profile, we determine exactly the voltage--charge relation in the transmission line where the front propagates and the squared local sound speed. The latter plays an important role in our analysis, determining the admissibility conditions for the front. We specifically consider fronts with a power-logistic profile and its special case, the hyperbolic-tangent profile. All physically admissible fronts of this form are shocks rather than kinks. For such shocks, the voltage--charge relation and the squared local sound speed--charge relation are expressed in terms of the lower incomplete beta function. We formulate the modified inverse-design method to calculate the profile of a shock propagating in the transmission line with a prescribed voltage--charge relation. We examine the local spectral properties of the front propagating in the transmission line in terms of the original partial differential equations and show that the short-wavelength perturbation modes are damped. The kink--shock distinction obtained from the mechanical analogy is independently confirmed through the long-wavelength characteristic analysis.

pdfLaTeX, 9+epsilon pages, 3 Figures. The manuscript has undergone major revision. Submitted for publication in Mathematics