self-similarity

These lectures were delivered to the high school students at the School – seminar «Nonlinear Days for Youth in Saratov – 2012» in October 2012. They present the two ways of historical investigation of symmetry. The first way is self-similarity, i.e. invariance at dimension scale changing. In a more general way the term «scaling» is used, meaning the existence of power-law correlation between some variable and variables x1, ...xn: y = Ax₁^α1...x_n^αn, where A, α₁,...α_n – are constant.

Self-organized structures after ion-beam irradiation in solid materials have been studied using the method of fractal dimension. General computer method of the scale invariance evaluation for exposed dispersive structures is described. It was demonstrated that structures after irradiation can be characterized by the compatibility of scale invariance properties at different scale levels.

In this work we study the dynamics of wave packets propagation of a few interacting quantum particles with different types of spatial inhomogeneity. Single particle or, equivalently, many noninteracting particles are localized in the case of spatial disorder, and experience localization–delocalization transition in the case of quasi-periodic inhomogeneity. In the other limiting case of many interacting particles, the problem is solved in the mean-field approximation, which leads to discrete nonlinear Schrodinger equation.