A Unified Explicit Construction of 2N-Soliton Solutions for Evolution Equations Determined by 2 × 2 AKNS System**The project supported by the State Key Program of Basic Research of China (Grant No. G1998030600) and the Natural Science Foundation of Shanghai, China (Grant No. ZD14012)

Abstract

An explicit N-fold Darboux transformation for evolution equations determined by general 2 × 2 AKNS system is constructed. By using the Darboux transformation, the solutions of the evolution equations are reduced to solving a linear algebraic system, from which a unified and explicit formulation of 2N-soliton solutions for the evolution equation are given. Furthermore, a reduction technique for MKdV equation is presented, and an N-fold Darboux transformation of MKdV hierarchy is constructed through the reduction technique. A Maple package which can entirely automatically output the exact N-soliton solutions of the MKdV equation is developed.