Hyperbolic-Tangent Shocks in a Lossy Nonlinear Transmission Line
We solve exactly the inverse problem for traveling fronts in a lossy nonlinear transmission line. Starting from a prescribed hyperbolic-tangent profile, we determine the voltage--charge relation that supports it. In the general case, the solution is expressed in terms of the lower incomplete beta function, while for positive integer or half-integer values of the front-width parameter $m$, it reduces to elementary functions. We analyze the distinction between kinks and shocks and show that all physically admissible fronts obtained in the exact construction are shocks. We obtain an approximate solution for the inverse problem which gives the voltage--charge relation in terms of elementary functions (for any $m$). We also solve the direct problem and obtain the approximate hyperbolic-tangent shock profile for a prescribed cubic voltage--charge relation. In the inverse approach, the shock parameters may be prescribed independently, subject to the admissibility conditions, whereas in the direct approach they are determined by the prescribed voltage--charge relation. The exact and approximate inverse solutions agree through order $m^{-2}$ in the broad-shock limit, and numerical comparisons show good agreement even outside this asymptotic regime.