Breathers and Q-Breathers: two sides of the same coin

We construct, and approximate from the continuum, two-parameter families of time periodic, small amplitude, localized solutions, for both the focusing and defocusing finite discrete nonlinear Schrödinger models, with Dirichlet boundary conditions. Within such families, depending on the parameters, both real space localization (breathers) and Fourier space localization (Q-breathers) are present. For the former type of solutions, convergence to the ground state of the focusing infinite chain is also proved; for the latter, a description of the localization properties is given, and some numerical results on the difference between the focusing and defocusing cases are explained. The proofs are based on continuation tools, ideas from the finite element methods, and techniques of convergence of variational problems.