The n-fold Darboux transformation for the Kundu-Eckhaus equation and dynamics of the smooth positon solutions

Abstract

In this paper, we completely discuss the derivation of Darboux transformation (DT) of the Kundu-Eckhaus (KE) equation by a direct way and an indirect way respectively. The compact determinant representation of formula of n-fold DT for the KE equation is constructed and nth-order solution is presented by the DT. Furthermore, we obtain the formula of the n-positon solution for the KE equation by using the Taylor expansion respect to degenerate eigenvalues λ2k−1→λ1(k=1,2,…,n). Especially, the exact expressions of soliton and positon solutions are obtained by the corresponding formulas. The dynamics of the smooth positons of the KE equation are discussed by using the decomposition of the modulus square, which can be used to depict approximately the trajectories and time-dependent ‘phase shifts’ of positons after the collision.