Formation of shock discontinuities for a class of nonlinear transmission lines

Abstract

We consider the problem of nonexistence of smooth globally defined solutions to a quasilinear hyperbolic system of equations; these equations are shown to model the behavior of a distributed parameter nonlinear transmission line, with voltage dependent capacitance and nonzero resistance and leakage conductance. Specific conditions are exhibited under which shocks form if the gradients of the initial data are, pointwise, sufficiently large. The analysis is based upon a study of the behavior of appropriate set of Riemann Invariants along their respective characteristic curves.