  We present a new generalization of the steepest descent method introduced by
Deift and Zhou for matrix Riemann-Hilbert problems and use it to study the
semiclassical limit of the focusing nonlinear Schroedinger equation with real
analytic, even, bell-shaped initial data. We provide explicit strong locally
uniform asymptotics for a sequence of exact solutions corresponding to initial
data that has been modified in an asymptotically small sense. We call this
sequence of exact solutions a semiclassical soliton ensemble. Our asymptotics
are valid in regions of the (x,t) plane where a certain scalar complex phase
function can be found. We characterize this complex phase function directly by
a finite-gap ansatz and also via the critical point theory of a certain
functional; the latter provides the correct generalization of the variational
principle exploited by Lax and Levermore in their study of the zero-dispersion
limit of the Korteweg-de Vries equation.
