Effective integration of the nonlinear vector Schrödinger equation

A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schrödinger equation (Manakov system) is developed. The allied spectral variety is a trigonal Riemann surface, which is described explicitly and the solutions of the equations are given in terms of θθ-functions of the surface. The final formulae are effective in the sense that all entries, like transcendental constants in exponentials, winding vectors etc., are expressed in terms of the prime-form of the curve and well algorithmized operations on them. That made the result available for direct calculations in applied problems implementing the Manakov system. The simplest solutions in Jacobian ϑϑ-functions are given as a particular case of general formulae and are discussed in detail.