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Advances in Mathematical Physics
Volume 2018 (2018), Article ID 1375653, 9 pages
Research Article

Generalized Matrix Exponential Solutions to the AKNS Hierarchy

1School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China
2Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA
3College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China
4Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
5College of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090, China
6Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

Correspondence should be addressed to Jian-bing Zhang

Received 4 December 2017; Revised 8 January 2018; Accepted 14 January 2018; Published 13 February 2018

Academic Editor: Zhijun Qiao

Copyright © 2018 Jian-bing Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Generalized matrix exponential solutions to the AKNS equation are obtained by the inverse scattering transformation (IST). The resulting solutions involve six matrices, which satisfy the coupled Sylvester equations. Several kinds of explicit solutions including soliton, complexiton, and Matveev solutions are deduced from the generalized matrix exponential solutions by choosing different kinds of the six involved matrices. Generalized matrix exponential solutions to a general integrable equation of the AKNS hierarchy are also derived. It is shown that the general equation and its matrix exponential solutions share the same linear structure.