Quantized Quasi-Two Dimensional Bose-Einstein Condensates with Spatially Modulated Nonlinearity

We investigate the localized nonlinear matter waves of the quasi-two dimensional Bose-Einstein condensates with spatially modulated nonlinearity in harmonic potential. It is shown that the whole Bose-Einstein condensates, similar to the linear harmonic oscillator, can have an arbitrary number of localized nonlinear matter waves with discrete energies, which are mathematically exact orthogonal solutions of the Gross-Pitaevskii equation. Their novel properties are determined by the principle quantum number n and secondary quantum number l: the parity of the matter wave functions and the corresponding energy levels depend only on n, and the numbers of density packets for each quantum state depend on both n and l which describe the topological properties of the atom packets. We also give an experimental protocol to observe these novel phenomena in future experiments.