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Aurones and their analogues are convenient twoatom units for the rapid construction of valuable spiroheterocycles. Here, we demonstrate that the Morita-Baylis-Hillman carbonates of isatins can be efficiently assembled with thioaurones, aurones, or even methyleneindolinones in a highly asymmetric [3 + 2] annulation manner, affording a broad spectrum of densely functionalized spiro-benzothiophene, -benzofuran and bisspirooxindole frameworks in high yields with good to excellent stereoselectivity.

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The modulational instability of two-component Bose–Einstein condensates (BECs) under an external parabolic potential is discussed. Based on the trapped two-component Gross–Pitaevskill equations, a time-dependent dispersion relation is obtained analytically by means of the modified lens-type transformation and linear stability analysis. It is shown that a modulational unstable time scale exists for trapped two-component BECs. The modulational properties—which are determined by the wave number, external trapping parameter, intra- and inter-species atomic interactions—are modified significantly. The analytical results are confirmed by direct numerical simulation. Our results provide a criterion for judging the occurrence of instability of the trapped two-component BECs in experiment.