Linear dissipative magnetohydrodynamics (MHD) shows that driven MHD waves in magnetic plasmas with high Reynolds number exhibit a near resonant behaviour if the frequency of the wave becomes equal to the local Alfve´n or slow frequency of a magnetic surface. This near resonant behaviour is confined to a thin dissipative layer which embraces the resonant magnetic surface. Although the driven MHD waves have small amplitudes far away from the resonant magnetic surface, this near-resonant behaviour in the dissipative layer may cause a breakdown of linear theory. In the present paper we deal with the nonlinear behaviour of driven MHD waves in the slow wave dissipative layer. The method of matched asymptotic expansions is used to obtain the nonlinear equation for wave variables inside the dissipative layer. The concept of connection formulae introduced into the theory of linear resonant MHD waves by Sakurai, Goossens, and Hollweg [Sol. Phys.133, 227 (1991)] is extended to include nonlinear effects in the dissipative layer for slow resonant waves. The absorption of the slow resonant wave in the dissipative layer generates a shear flow parallel to the magnetic surfaces with a characteristic velocity of the order of&egr;1/2,where &egr; is the dimensionless amplitude of perturbations far away from the dissipative layer. ©1997 American Institute of Physics.