# Nonlinear Quantum Graphs: Canonical Perturbation Approach to Stationary States

**Speaker:** Sven Gnutzmann, School of Mathematical Sciences

**Title:** Nonlinear Quantum Graphs: Canonical Perturbation Approach to Stationary States

**Abstract:** I consider nonlinear SchrÃ¶dinger equations on metric graphs as a qualitative model for nonlinear fibre-optical networks or Bose-Einstein condensates in quasi-one-dimensional traps. As exact wave solutions along each edge are available (at least for the simplest case of a cubic non-linearity) the problem reduces to a set of coupled nonlinear algebraic equations that describe the matching conditions at the nodes of the graph. While these equations are often not very illuminating numerical solutions show a large variety of nonlinear effects such as bifurcations or multistability.

Using canonical perturbation theory I present approximate wave solutions along the edges that simplify the analysis. The perturbative equations remain valid in the short-wavelength asymptotics. In this regime they allow for sufficiently large intensities that allow for genuine non-linear effects to be studied.

The talk will focus on the basic framework and illustrative examples.