On a Nonlinear Hyperbolic Equation Describing Transmission Lines, Cell Movement, and Branching Random Walks

Abstract

This is a preliminary and expository report on a nonlinear hyperbolic equation that arises from a variety of distinct phenomena. We derive the equation

$$ {\varepsilon ^2}{v_{tt}} + \left( {1 + g\left( v \right)} \right){v_t} = {k^2}{v_{xx}} + f\left( v \right) $$

((1))

as the equation for the voltage along a transmission line with nonlinear shunt conductance and a series inductance along the length of the line, from simple models of movement and reproduction in tissue cells and one celled organisms, and from a mathematical treatment of a branching random walk. In addition, with the proper scaling and choice of f and g this nonlinear hyperbolic equation can be viewed as the equation that describes a continuum of coupled van der Pol oscillators. In equation (1) the value of ε ² need not be small, but the choice of the notation ε ² suggests analogies with other well known nonlinear partial differential equations, and we will mention some of these analogies below. The purpose of this report is to briefly explain and motivate all of these derivations and to present some basic results about the solutions of this equation. In addition, the purpose is to show how the probabilistic interpretation of the equation arising out of the branching random walk helps in the understanding and motivation of the results. Detailed proofs of the new results will be presented elsewhere.

Supported in part by NSF Grant DMS-8301840

Supported in part by NSF Grant DMS-8301840 and NIH Grant GM-29123