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We study the periodic traveling wave solutions of the derivative nonlinearSchr\"{o}dinger equation (DNLS). It is known that DNLS has two types ofsolitons on the whole line; one has exponential decay and the other hasalgebraic decay. The latter corresponds to the soliton for the massless case.In the new global results recently obtained by Fukaya, Hayashi and Inui, theproperties of two-parameter of the solitons are essentially used in the proof,and especially the soliton for the massless case plays an important role. Toinvestigate further properties of the solitons, we construct exact periodictraveling wave solutions which yield the solitons on the whole line includingthe massless case in the long-period limit. Moreover, we study the regularityof the convergence of these exact solutions in the long-period limit.Throughout the paper, the theory of elliptic functions and elliptic integralsis used in the calculation.

Long-period limit of exact periodic traveling wave solutions for the derivative nonlinear Schr\"{o}dinger equation