10 research outputs found
Stationary Nonlinear Schr\"odinger Equation on Simplest Graphs: Boundary conditions and exact solutions
We treat the stationary (cubic) nonlinear Schr\"odinger equation (NSLE) on
simplest graphs. Formulation of the problem and exact analytical solutions of
NLSE are presented for star graphs consisting of three bonds. It is shown that
the method can be extended for the case of arbitrary number of bonds of star
graphs and for other simplest topologies such as tree and loop graphs. The case
of repulsive and attractive nonlinearities are treated separately
Transport in simple networks described by integrable discrete nonlinear Schr\"Aodinger equation
We elucidate the case in which the Ablowitz-Ladik (AL) type discrete
nonlinear Schr\"Aodinger equa- tion (NLSE) on simple networks (e.g., star
graphs and tree graphs) becomes completely integrable just as in the case of a
simple 1-dimensional (1-d) 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 1-d discrete chain, but is multiplied by the inverse of
square root of bond-dependent nonlinearity; (2) nonlinearities at individual
bonds 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 practical
issue, with use of AL soliton injected through the incoming bond, we obtain
transmission probabilities inversely proportional to the strength of
nonlinearity on the outgoing bonds