Shooting method for nonlinear singularly perturbed boundary-value problems

Asymptotic formulas, as epsilon --> 0, are derived for the solutions of the nonlinear differential equation epsilonu" + Q(u) = 0 with boundary conditions u(1) = u(1) = 0 or u'(1) = u'(1) = 0. The nonlinear term Q(u) behaves like a cubic; it vanishes at s(-), 0, s(+) and nowhere else in [s(-), s(+)], where s(-) < 0 < s(+). Furthermore, Q'(s(+/-)) < 0, Q'(0) > 0 and the integral of Q on the interval [s(-), s(+)] is zero. Solutions to these boundary-value problems are shown to exhibit internal shock layers, and the error terms in the asymptotic approximations are demonstrated to be exponentially small. Estimates are obtained for the number of internal shocks that a solution can have, and the total numbers of solutions to these problems are also given. All results here are established rigorously in the mathematical sense.