  The inverse scattering transform for the focusing nonlinear Schrodinger
equation is presented for a general class of initial conditions whose
asymptotic behavior at infinity consists of counterpropagating waves. The
formulation takes into account the branched nature of the two asymptotic
eigenvalues of the associated scattering problem. The Jost eigenfunctions and
scattering coefficients are defined explicitly as single-valued functions on
the complex plane with jump discontinuities along certain branch cuts. The
analyticity properties, symmetries, discrete spectrum, asymptotics and behavior
at the branch points are discussed explicitly. The inverse problem is
formulated as a matrix Riemann-Hilbert problem with poles. Reductions to all
cases previously discussed in the literature are explicitly discussed. The
scattering data associated to a few special cases consisting of physically
relevant Riemann problems are explicitly computed.
