  In this paper a novel numerical scheme for finding the sparse self-localized
states of a nonlinear system of equations with missing spectral data is
introduced. As in the Petviashivili's and the spectral renormalization method,
the governing equation is transformed into Fourier domain, but the iterations
are performed for far fewer number of spectral components (M) than classical
versions of the these methods with higher number of spectral components (N).
After the converge criteria is achieved for M components, N component signal is
reconstructed from M components by using the l1 minimization technique of the
compressive sampling. This method can be named as compressive spectral
renormalization (CSRM) method. The main advantage of the CSRM is that, it is
capable of finding the sparse self-localized states of the evolution
equation(s) with many spectral data missing.
