Abstract. The n-fold Darboux transformation (DT) is a 22 matrix for the Kaup-Newell (KN) system. In this paper, each element of this matrix is expressed by a ratio of (n + 1) × (n + 1) determinant and n × n determinant of eigenfunctions. Using these formulae, the expressions of the q[n] and r [n] in KN system are generated by n-fold DT. Further, under the reduction condition, the rogue wave,rational traveling solution, dark soliton, bright soliton, breather solution, periodic solution of the derivative nonlinear Schrödinger(DNLS) equation are given explicitly by different seed solutions. In particular, the rogue wave and rational traveling solution are two kinds of new solutions. The complete classification of these solutions generated by one-fold DT is given in the table on page.

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In this paper, we establish some important properties of M-tensors. We derive upper and lower bounds for the minimum eigenvalue of M-tensors, bounds for eigenvalues of M-tensors except the minimum eigenvalue are also presented; finally, we give the Ky Fan theorem for M-tensors. MSC: 15A18; 15A69; 65F15; 65F10

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