Multifractals in Soliton sea on Fibonacci Lattice

Systems exhibiting Bose-Einstein condensation are suitable for fabricating artificially designed structures, e.g., the bichromatic potentials have attracted much attention. More exotic structures also appear to be experimentally feasible in the optical imaging system with high resolution. We consider one of exotic structures, Fibonacci potential, which is quasiperiodic. On the Fibonacci potential without nonlinear interaction, it is known that the spectrum is singular continuous and all the eigenvectors are called a critical state, in which multifractal states are included. On the other hand, the physical properties of the Bose-Einstein condensates confined in a optical lattice can be described in terms of the Schr\"odinger equation with nonlinear term via particle-particle interactions. We numerically and mathematically investigated nonlinear Schr\"odinger equation on Fibonacci potential focusing on the competition between nonlinear fluctuation and criticality~[M.~Takahashi {\it et al.}, arXiv:1110.6328]. The conclusion is that the critical states with the spectrum in the Cantor set retains their profile irrespective of the strength of the nonlinearity. The spectrum for the critical states is in a sea of ``stationary solitons" which appear as a result of nonlinear effects.