The solutions q [n] generated from a periodic "seed" q = ce i(as+bt) of the nonlinear Schrö-dinger (NLS) by n-fold Darboux transformation is represented by determinant. Furthermore, the s-periodic solution and t-periodic solution are given explicitly by using q [1] . The curves and surfaces (F 1 , F 2 , F 3 ) associated with q [n] are given by means of Sym formula. Meanwhile, we show periodic and asymptotic properties of these curves.
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