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We study the stationary states for the nonlinear Schr\"odinger equation onthe Fibonacci lattice which is expected to be realized by Bose-Einsteincondensates loaded into an optical lattice. When the model does not have anonlinear term, the wavefunctions and the spectrum are known to show fractalstructures. Such wavefunctions are called critical. We present a phase diagramof the energy spectrum for varying the nonlinearity. It consists of threeportions, a forbidden region, the spectrum of critical states, and the spectrumof stationary solitons. We show that the energy spectrum of critical statesremains intact irrespective of the nonlinearity in the sea of a large number ofstationary solitons.

Multifractals Competing with Solitons on Fibonacci Optical Lattice