Because of the obvious advantage in long time predictions it is useful to convert dynamical problems of flows into problems involving maps. For Hamiltonian flows this in effect is equivalent to identifying an area preserving map in the Poincare surface of section. The preservation of canonical structure of the Hamiltonian flow in the surface of section can lead to a description in terms of discrete canonical equations in the surface of section. This property is utilized here to convert the Hamiltonian flow problem of the dynamic evolution of self-focusing of an electromagnetic beam (width/phase front curvature dynamics) with beam propagation distance into an equivalent mapping problem for a wide range of initial conditions. The nonlinear Schrodinger equation is thereby converted to a map in a restricted sense.
"Map for self-focusing of a laser beam in a nonlinear medium", Proc. SPIE 2039, Chaos in Optics, (1 December 1993); doi: 10.1117/12.165082; https://doi.org/10.1117/12.165082