Exact solutions and residual symmetries of the Ablowitz–Kaup–Newell–Segur system**Project supported by the National Natural Science Foundation of China (Grant Nos. 11305031, 11365017, and 11305106), the Natural Science Foundation of Guangdong Province, China (Grant No. S2013010011546), the Natural Science Foundation of Zhejiang Province, China (Grant No. LQ13A050001), the Science and Technology Project Foundation of Zhongshan, China (Grant Nos. 2013A3FC0264 and 2013A3FC0334),

Abstract

The residual symmetries of the Ablowitz–Kaup–Newell–Segur (AKNS) equations are obtained by the truncated Painlevé analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symmetries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calculation on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space–time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Bäcklund transformations between the AKNS equations and the Schwarzian AKNS equation.