Nonlocal nonlinear Schrödinger equation on metric graphs: A model for generation and transport of parity-time-symmetric nonlocal solitons in networks.

We consider the parity-time (PT)-symmetric, nonlocal, nonlinear Schrödinger equation on metric graphs. Vertex boundary conditions are derived from the conservation laws. Soliton solutions are obtained for the simplest graph topologies, such as star and tree graphs. The integrability of the problem is shown by proving the existence of an infinite number of conservation laws. A model for soliton generation in such PT-symmetric optical fibers and their networks governed by the nonlocal nonlinear Schrödinger equation is proposed. Exact formulas for the number of generated solitons are derived for the cases when the problem is integrable. Numerical solutions are obtained for the case when integrability is broken.

Dirac particles on periodic quantum graphs.

We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasiperiodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum graphs is obtained. Band spectra of the periodic quantum graphs of different topologies are calculated. Universality of the probability to be in the spectrum for certain graph topologies is observed.

Reflectionless propagation of Manakov solitons on a line: A model based on the concept of transparent boundary conditions.

We consider the problem of the absence of backscattering in the transport of Manakov solitons on a line. The concept of transparent boundary conditions is used for modeling the reflectionless propagation of Manakov vector solitons in a one-dimensional domain. Artificial boundary conditions that ensure the absence of backscattering are derived and their numerical implementation is demonstrated.

Dirac particles in transparent quantum graphs: Tunable transport of relativistic quasiparticles in branched structures.

We consider the dynamics of relativistic spin-half particles in quantum graphs with transparent branching points. The system is modeled by combining the quantum graph concept with the one of transparent boundary conditions applied to the Dirac equation on metric graphs. Within such an approach, we derive simple constraints, which turn the usual Kirchhoff-type boundary conditions at the vertex equivalent to the transparent ones. Our method is applied to quantum star graph. An extension to more complicated graph topologies is straightforward.

Transparent nonlinear networks.

We consider the reflectionless transport of solitons in networks. The system is modeled in terms of the nonlinear Schrödinger equation on metric graphs, for which transparent boundary conditions at the branching points are imposed. This approach allows to derive simple constraints, which link the equivalent usual Kirchhoff-type vertex conditions to the transparent ones. Our method is applied to a metric star graph. An extension to more complicated graph topologies is straightforward.

[Autoimmune hemolytic anemia associated with mesenteric teratoma]. / Autoimmunnaia gemoliticheskaia anemiia, assotsiirovannaia s teratomoi bryzheiki tonkoi kishki.

The paper describes a case of autoimmune hemolytic anemia (AIHA) in a 27-year-old woman whose examination revealed mesenteric teratoma. AIHA was characterized by a hypertensive crisis and a temporary response to corticosteroid therapy that was complicated by the development of somatogenic psychosis and discontinued. A relapse of hemolysis developed 6 months later. The patient underwent laparoscopic splenectomy and removal of mesenteric root teratoma. Immediately after surgery, a hematological response was obtained as relief of hemolysis and restoration of a normal hemoglobin level. There is a sustained remission of AIHA for the next 16 months.

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.