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We prove existence of a special class of solutions to the (elliptic)Nonlinear Schroeodinger Equation $- \epsilon^2 \Delta \psi + V(x) \psi =|\psi|^{p-1} \psi$, on a manifold or in the Euclidean space. Here V representsthe potential, p an exponent greater than 1 and $\epsilon$ a small parametercorresponding to the Planck constant. As $\epsilon$ tends to zero (namely inthe semiclassical limit) we prove existence of complex-valued solutions whichconcentrate along closed curves, and whose phase is highly oscillatory.Physically, these solutions carry quantum-mechanical momentum along the limitcurves. In this first part we provide the characterization of the limit set,with natural stationarity and non-degeneracy conditions. We then construct anapproximate solution up to order $\epsilon^2$, showing that these conditionsappear naturally in a Taylor expansion of the equation in powers of $\epsilon$.Based on these, an existence result will be proved in the second part.

Solutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions