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We elucidate the case in which the Ablowitz-Ladik (AL) type discretenonlinear Schr\"Aodinger equa- tion (NLSE) on simple networks (e.g., stargraphs and tree graphs) becomes completely integrable just as in the case of asimple 1-dimensional (1-d) discrete chain. The strength of cubic nonlinearityis different from bond to bond, and networks are assumed to have at least twosemi-infinite bonds with one of them working as an incoming bond. The presentwork is a nontrivial extension of our preceding one (Sobirov et al, Phys. Rev.E 81, 066602 (2010)) on the continuum NLSE to the discrete case. We find: (1)the solution on each bond is a part of the universal (bond-independent) ALsoliton solution on the 1-d discrete chain, but is multiplied by the inverse ofsquare root of bond-dependent nonlinearity; (2) nonlinearities at individualbonds around each vertex must satisfy a sum rule; (3) under findings (1) and(2), there exist an infinite number of constants of motion. As a practicalissue, with use of AL soliton injected through the incoming bond, we obtaintransmission probabilities inversely proportional to the strength ofnonlinearity on the outgoing bonds.

Transport in simple networks described by integrable discrete nonlinear Schr\"Aodinger equation