Effective integration of the nonlinear vector Schrِdinger 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 θ -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 ϑ -functions are given as a particular case of general formulae and are discussed in detail.