Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation.

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.

Ideal quantum gas in an expanding cavity: nature of nonadiabatic force.

We consider a quantum gas of noninteracting particles confined in the expanding cavity and investigate the nature of the nonadiabatic force which is generated from the gas and acts on the cavity wall. First, with use of the time-dependent canonical transformation, which transforms the expanding cavity to the nonexpanding one, we can define the force operator. Second, applying the perturbative theory, which works when the cavity wall begins to move at time origin, we find that the nonadiabatic force is quadratic in the wall velocity and thereby does not break the time-reversal symmetry, in contrast with general belief. Finally, using an assembly of the transitionless quantum states, we obtain the nonadiabatic force exactly. The exact result justifies the validity of both the definition of the force operator and the issue of the perturbative theory. The mysterious mechanism of nonadiabatic transition with the use of transitionless quantum states is also explained. The study is done for both cases of the hard- and soft-wall confinement with the time-dependent confining length.

Integrable nonlinear Schrödinger equation on simple networks: connection formula at vertices.

We study the case in which the nonlinear Schrödinger equation (NLSE) on simple networks consisting of vertices and bonds has an infinite number of constants of motion and becomes completely integrable just as in the case of a simple one-dimensional (1D) chain. Here 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 connection formula at vertices obtained from norm and energy conservation rules shows (1) the solution on each bond is a part of the universal (bond-independent) soliton solution of the completely integrable NLSE on the 1D 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. Under these conditions, we also showed an infinite number of constants of motion. The argument on a branched chain or a primary star graph is generalized to other graphs, i.e., general star graphs, tree graphs, loop graphs and their combinations. As a relevant issue, with use of reflectionless propagation of Zakharov-Shabat's soliton through networks we have obtained the transmission probabilities on the outgoing bonds, which are inversely proportional to the bond-dependent strength of nonlinearity. Numerical evidence is also given to verify the prediction.