title: Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation.
abstract: We elucidate the case in which the Ablowitz-Ladik (AL)-type discrete nonlinear Schrödinger equation (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple one-dimensional (1D) discrete chain. The strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The present work 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) AL soliton solution on the 1D discrete chain, but it is multiplied by the inverse of the square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule; and (3) under findings 1 and 2, there exist an infinite number of constants of motion. As a practical issue, with the use of an AL soliton injected through the incoming bond, we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds.
authors.affiliation: Faculty of Physics, National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan.
authors.forename: K
authors.initials: K
authors.lastname: Nakamura
authors.forename: Z A
authors.initials: ZA
authors.lastname: Sobirov
authors.forename: D U
authors.initials: DU
authors.lastname: Matrasulov
authors.forename: S
authors.initials: S
authors.lastname: Sawada
mesh_terms: D009038:Motion; D017711:Nonlinear Dynamics