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We construct harmonic maps into SU(1,1)/U(1) satrting from Smyth potentials\xi, by the DPW method, In this method, harmonic maps are obtained from theIwasawa factorization of a solution L of L^{-1} dL = \xi. However, the Iwasawafactorization in the case of a noncompact group is not always global. We showthat L can be expressed in terms of Bessel functions and from the asymptoticexpansion of Bessel functions we solve a Riemann-Hilbert problem to give aglobal Iwasawa factorization. In this way we give a more direct proof than inthe work of Dorfmeister-Guest-Rossman (2010), while avoiding the generalisomonodromy theory used by Guest-Its-Lin (2015).

Globality of the DPW construction for Smyth potentials in the case of SU(1,1)