2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/80530This is the first of a series of papers devoted to the study of classical initial-boundary value problems of Dirichlet, Neumann and mixed type for the Nonlinear Schrödinger equation on the segment. Considering proper periodic discontinuous extensions of the profile, generated by suitable point-like sources, we show that the above boundary value problems can be rewritten as nonlinear dynamical systems for suitable sets of algebro-geometric spectral data, generalizing the classical Dubrovin equations. In this paper we consider, as a first illustration of the above method, the case of the Dirichlet problem on the segment with zero-boundary value at one end, and we show that the corresponding dynamical system for the spectral data can be written as a system of ODEs with algebraic right-hand side.29 pages, LaTeX, 2 Encapsulated Postscript figuresExactly Solvable and Integrable SystemsHigh Energy Physics - TheoryMathematical PhysicsAlgebraic GeometryAnalysis of PDEstext