Sven Gnutzmann

Nonlinear Quantum Graphs: Canonical Perturbation Approach to Stationary Waves

I will consider the stationary nonlinear Schrödinger equation on a metric graph. This is a model for understanding the interplay between nonlinearity and connectivity, e.g. in a network of optical fibres or a Bose-Einstein condensate in a trap of non-trivial topology. Formally the solution of the differential equation with matching conditions at the vertices may be reduced to a set of nonlinear coupled algebraic equations for a finite number of parameters. While this may be solved numerically we will show that some analytical results may be obtained using the appropriate canonical perturbation theory of Hamilonian dynamics.

The emergence of various asymptotic regimes (such as small wavelength or low intensities) is discussed and approximate solutions for some basic non-trivial metric graphs are presented.

If time allows some recent results on the characterisation of solutions on star graphs in term of the nodal structure will be presented.