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In the present work, we establish the approximation of nonlinear stochasticpartial differential equation (SPDE) driven by cylindrical {\alpha}-stableL\'evy processes via modulation or amplitude equations.  We study SPDEs with a cubic nonlinearity, where the deterministic equation isclose to a change of stability of the trivial solution. The natural separationof time-scales close to this bifurcation allows us to obtain an amplitudeequation describing the essential dynamics of the bifurcating pattern, thusreducing the original infinite dimensional dynamics to a simplerfinite-dimensional effective dynamics. In the presence of a multiplicativestable L\'evy noise that preserves the constant trivial solution we study theimpact of noise on the approximation.  In contrast to Gaussian noise, where non-dominant pattern are uniformly smallin time due to averaging effects, large jumps in the L\'evy noise might lead tolarge error terms, and thus new estimates are needed to take this into account.

Modulation and amplitude equations on bounded domains for nonlinear SPDEs driven by cylindrical {\alpha}-stable L\'evy processes