1. Stationary scattering on non-linear networks (qm-graphs) Sven Gnutzmann (Nottingham), Stas Derevyanko (Aston) and Uzy Smilansky TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA Abstract: Transmission through a complex network of nonlinear one-dimensional leads is discussed by extending the stationary scattering theory on quantum graphs to the nonlinear regime. The resonances which dominate linear scattering are shown to be extremely sensitive to the nonlinearity and display multi-stability and hysteresis. This work provides a framework for the study of light propagation in complex optical networks, and for studying universal properties of Bose-Einstein Condensate (BEC) in connected chaotic traps. PHYS. REV. A 83, 033831 (2011) Warsaw May, 2011

2. Motivations: 1.Develop a paradigm for studying the effects of non linearity in physical systems. 2.Applications for fiber lasers consisting of inter connected network of optical fibers. Introduction : Scattering on linear quantum graphs

3. Example:

4. Amplification of the interior wave-function at resonance k

5. Nonlinear Schroedinger equation on metric graphs – stationary solutions. In this way the NLSE can be written as Euler-Lagrange equations for a particle in two dimensions with a central potential and polar coordinates x

6. Classical particle in an axially symmetric potential with two constants of motion H and l Elliptic integrals: manageable but not “user-friendly”

7. attractive repulsive Explicit solution on the line

8. NLSE on graphsGraph: V vertices, B internal bonds, N external leads Linear wave equation on the external leads Nonlinear wave equation on the internal bonds From the classical picture on the bonds: Conservation of H and of Boundary (Matching) condition at the vertex : 1.Continuity: 2. Flux conservation:

9. Using this formalism one can easily prove that scattering from the graph conserves flux: However and only in the linear limit : The nonlinear scattering operator is a norm preserving mapping

10. Scattering from a star graph

11. Back to a more complex graph

12. Numerical simulations for the graph

13. Conclusions: 1.Nonlinear effects are magnified near resonances, the narrower the resonance the stronger the magnification. 2.Graphs are cavities which support narrow resonances and therefore are more susceptible to nonlinear effects. 3.This may have important effect on e.g., fiber-optic lasers. Open problems: 0. Linear theory: statistics of the amplification factor. 1.The non-linear scattering operator may have interesting properties which are not yet investigated. 2.Connection to time dependent theory. 3.Physical basis of the boundary conditions. 4.The applicability of the NLSE for the description of optical fibers. 5. Inclusion of dissipation, optically active materials… תודה רבה