Hyperbolic-Tangent Shocks in a Lossy Nonlinear Transmission Line

We investigate traveling fronts in a lossy nonlinear transmission line. By prescribing a hyperbolic-tangent profile with dimensionless width $m$, we solve the inverse problem of determining the voltage--charge relation for which this profile is an exact solution. The exact relation is expressed in terms of the Gauss hypergeometric function. We also construct an approximate solution by assuming a cubic voltage--charge relation and replacing the resulting quadratic damping function with its optimal linear least-squares approximation. This construction determines the cubic coefficients and yields a modified governing equation that admits the prescribed profile as an exact solution. In both constructions, physical admissibility selects a shock front. In the broad-front limit, the approximate normalized force agrees with the exact result through order $m^{-2}$; numerical comparisons show that the agreement remains good even outside the asymptotic region of large $m$. Finally, a characteristic analysis of the original partial differential equations distinguishes compressive shocks from noncompressive kinks without recourse to the mechanical analogy.