Standing waves and homoclinic solutions for periodic discrete Schrödinger systems with saturable nonlinearities

In this study, we obtained the existence of non-trivial standing waves and homoclinic solutions for periodic discrete nonlinear Schrödinger (DNS) systems with nonlinearities being saturable (asymptotically linear) at infinity. The main innovations of this study are as follows. (1) In contrast to the usual approach, instead of starting from a sequence of approximate critical points of the true functional, we look for the sequence of true critical points of nearby functionals. (2) A ‘necessary and sufficient’ condition of the existence of non-trivial homoclinic solutions for DNS systems is obtained. (3) Our method can be applied to discrete Vinetskii–Kukhtarev (DVK) systems and equations. (4) Even in some special cases of our system, our results greatly extend the results of (Zhou & Yu 2010 J. Differ. Eqn. 249 , 1199–1212 ( doi:10.1016/j.jde.2010.03.010 )) and (Pankov & Rothos 2008 Proc. R. Soc. A 464 , 3219–3236 ( doi:10.1098/rspa.2008.0255 )).