Abstract:  Based on the Lagrangian action density under Born-Infeld type dynamics and
motivated by the one-dimensional prescribed mean curvature equation, we investigate the
polynomial function model in Born-Infeld theory in this paper with the form of
−([1−a(ϕ')² ]ϕ' )' =λf(ϕ(x)),
where λ> 0 is a real parameter, f ∈C² (0 , + ∞ ) is a nonlinear function. We are interested
in the exact number of positive solutions of the above nonlinear equation. We specifically
develop for the problem combined with a careful analysis of a time-map method.

Key words:  Time-map analysis, Exact number of solutions, Lagrangian action density; The polynomial function model, Born-Infeld theory