We propose a theoretical scheme to show the possibility of generating motional nonlinear coherent states and their superposition for an undamped vibrating micromechanical membrane inside an optical cavity. The scheme is based on an intensity-dependent coupling of the membrane to the radiation pressure field. We show that if the cavity field is initially prepared in a Fock state, the motional state of the membrane may evolve from vacuum state to a special type of nonlinear coherent states. By examining the nonclassical properties of the generated state of the membrane, including the quadrature squeezing and the sub-Poissonian statistics, we find that by varying the Lamb-Dicke parameter and the membrane's reflectivity one can effectively control those properties. In addition, the scheme offers the possibility of generating various types of the so-called nonlinear multicomponent Schrodinger cat sates of the membrane. We also examine the effect of the damping of the cavity field on the motional state of the membrane
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