摘要
We introduce two operator commutators by using different-degree loop algebras of the Lie algebra A1,then under the framework of zero curvature equations we generate two(2+1)-dimensional integrable hierarchies, including the(2+1)-dimensional shallow water wave(SWW) hierarchy and the(2+1)-dimensional Kaup–Newell(KN)hierarchy. Through reduction of the(2+1)-dimensional hierarchies, we get a(2+1)-dimensional SWW equation and a(2+1)-dimensional KN equation. Furthermore, we obtain two Darboux transformations of the(2+1)-dimensional SWW equation. Similarly, the Darboux transformations of the(2+1)-dimensional KN equation could be deduced. Finally,with the help of the spatial spectral matrix of SWW hierarchy, we generate a(2+1) heat equation and a(2+1) nonlinear generalized SWW system containing inverse operators with respect to the variables x and y by using a reduction spectral problem from the self-dual Yang–Mills equations.
基金
Supported by the National Natural Science Foundation of China under Grant No.11371361
the Shandong Provincial Natural Science Foundation of China under Grant Nos.ZR2012AQ011,ZR2013AL016,ZR2015EM042
National Social Science Foundation of China under Grant No.13BJY026
the Development of Science and Technology Project under Grant No.2015NS1048
A Project of Shandong Province Higher Educational Science and Technology Program under Grant No.J14LI58
