ON THREE-WAVE INTERACTION SCHRODINGER SYSTEMS WITH QUADRATIC NONLINEARITIES: GLOBAL WELL-POSEDNESS AND STANDING WAVES

Reported here are results concerning the global well-posedness in the energy space and existence and stability of standing-wave solutions for 1-dimensional three-component systems of nonlinear Schrodinger equations with quadratic nonlinearities. For two particular systems we are interested in, the global well-posedness is established in view of the a priori bounds for the local solutions. The standing waves are explicitly obtained and their spectral stability is studied in the context of Hamiltonian systems. For more general Hamiltonian systems, the existence of standing waves is accomplished with a variational approach based on the Mountain Pass Theorem. Uniqueness results are also provided in some very particular cases.