Stationary solutions and self-trapping in discrete quadratic nonlinear systems

We consider the simplest equations describing coupled quadratic nonlinear $\left({\chi }^{\mathrm{}\left(2\right)\mathrm{}}\right)$ systems, which each consists of a fundamental mode resonantly interacting with its second harmonic. Such discrete equations apply, e.g., to optics, where they can describe arrays of ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ waveguides, and to solid state physics, where they can describe nonlinear interface waves under the conditions of Fermi resonance of the adjacent crystals. Focusing on the monomer and dimer we discuss their Hamiltonian structure and find all stationary solutions and their stability properties. In one limit the nonintegrable dimer reduce to the discrete nonlinear Schrödinger (DNLS) equation with two degrees of freedom, which is integrable. We show how the stationary solutions to the two systems correspond to each other and how the self-trapped DNLS solutions gradually develop chaotic dynamics in the ${\chi }^{\mathrm{}\left(2\right)\mathrm{}}$ system, when going away from the near integrable limit.