  In the approximation of linear dissipative magnetohydrodynamics (MHD) it can
be shown 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 Alfven (or slow) frequency of a magnetic surface. This near resonant
behaviour is confined to a thin region, known as the dissipative layer, which
embraces the resonant magnetic surface. Although driven MHD waves have small
dimensionless amplitude far away from the resonant surface, this near-resonant
behaviour in the dissipative layer may cause a breakdown of linear theory. Our
aim is to study the nonlinear effects in the Alfven dissipative layer. In the
present paper, the method of simplified matched asymptotic expansions developed
for nonlinear slow resonant waves is used to describe nonlinear effects inside
the Alfven dissipative layer. The nonlinear corrections to resonant waves in
the Alfven dissipative layer are derived and it is proved that at the Alfven
resonance (with isotropic/anisotropic dissipation) wave dynamics can be
described by the linear theory with great accuracy.
