Summary: Plasmas are believed to be possibly the most abundant form of visible matter in the Universe. Investigation in this paper is given to a variable-coefficient derivative nonlinear Schrödinger equation describing the nonlinear Alfvén waves in an inhomogeneous plasma. With respect to the left polarized Alfvén wave, the odd-fold Darboux transformation, breather, rogue-wave and semirational solutions are constructed on the periodic background. Conditions for the antidark/gray/black soliton solutions or periodic solutions are obtained. Period of the Akhmediev breather is independent of the dispersion coefficient. When such an equation has a constant dispersion, with the increasing value of the dispersion coefficient, quasi-period of the Kuznetsov-Ma breather decreases, quasi-periods of the spatio-temporal breathers along the \(\xi\) and \(\tau\) axes both decrease and range of the rogue wave along the \(\tau\) axis decreases, where \(\xi\) and \(\tau\) are the stretched time and space variables, respectively. When such an equation has a \(\tau\)-dependent linear dispersion, with the increasing value of \(|\tau|\), quasi-period of the Kuznetsov-Ma breather decreases and quasi-periods of the spatio-temporal breathers along the \(\xi\) and \(\tau\) axes both decrease. When such an equation has a constant loss/gain, linearly periodic background is exhibited. When such an equation has a \(\tau\)-dependent linear loss/gain, parabolic-periodic background is shown.
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