Abstract
For the integrable couplings of Ablowitz-Kaup-Newell-Segur (ICAKNS) equations, N-fold Darboux transformation (DT) T-N, which is a 4 x 4 matrix, is constructed in this paper. Each element of this matrix is expressed by a ratio of the (4N + 1)-order determinant and 4N -order determinant of eigenfunctions. By making use of these formulae, the determinant expressions of N-transformed new solutions p([N]), q([N]) r([N]) and s([N]) are generated by this N-fold DT. Furthermore, when the reduced conditions q = -p* and s = -r* are chosen, we obtain determinant representations of N-fold DT and N-transformed solutions for the integrable couplings of nonlinear Schrodinger (ICNLS) equations. Starting from the zero seed solutions, one-soliton solutions are explicitly given as an example.