Existence of localized solutions in the parametrically driven and damped DNLS equation in high-dimensional lattices

Instead of using the homoclinic orbit approach, which was commonly taken when studying the localized solutions of the discrete non-linear Schrödinger (DNLS) equation in one-dimensional lattices, we apply the continuation theorem to investigate the existence, stability, and spatial complexity of the localized solutions, including bright breathers, dark breathers, and anti-phase breathers, of the parametrically driven and damped DNLS equation in high-dimensional lattices. In particular, we prove that the sufficient condition that the driving strength exceeds the damping constant is necessary for the system with weak coupling to possess localized solutions.