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General breather solution to the Sasa–Satsuma equation (SSE) is systematically investigated in this paper. We firstly transform the SSE into a set of three Hirota bilinear equations under a proper plane wave boundary condition. Starting from a specially arranged tau-function of the Kadomtsev–Petviashvili hierarchy and a set of 11 bilinear equations satisfied, we implement a series steps of reduction procedure, i.e. C-type reduction, dimension reduction and complex conjugate reduction, and reduce these 11 equations to three bilinear equations for the SSE. Meanwhile, the general breather solution to the SSE is found in determinant of even order. The one- and two-breather solutions are calculated and analysed in detail. 

This study describes general methods for the enantioselective syntheses of pharmaceutically relevant 1-aryl-2-heteroaryl- and 1,2-diheteroarylcyclopropane-1-carboxylates through dirhodium tetracarboxylate-catalysed asymmetric cyclopropanation of vinyl heterocycles with aryl- or heteroaryldiazoacetates. The reactions are highly diastereoselective and high asymmetric induction could be achieved using either ( R )-pantolactone as a chiral auxiliary or chiral dirhodium tetracarboxylate catalysts. For meta - or para -substituted aryl- or heteroaryldiazoacetates the optimum catalyst was Rh 2 ( R-p -Ph-TPCP) 4 . In the case of ortho -substituted aryl- or heteroaryldiazoacetates, the optimum catalyst was Rh 2 ( R -TPPTTL) 4 . For a highly enantioselective reaction with the ortho -substituted substrates, 2-chloropyridine was required as an additive in the presence of either 4 Å molecular sieves or 1,1,1,3,3,3-hexafluoroisopropanol (HFIP). Under the optimized conditions, the cyclopropanation could be conducted in the presence of a variety of heterocycles, such as pyridines, pyrazines, quinolines, indoles, oxadiazoles, thiophenes and pyrazoles.