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In this paper we consider nonlinear Schrodinger systems with periodicboundary condition in high dimension. We establish an abstract infinitedimensional KAM theorem and apply it to the nonlinear Schrodinger equationsystems with real Fourier Multiplier. By establishing a block-diagonal normalform, We prove the existence of a class of Whitney smooth small amplitudequasi-periodic solutions corresponding to finite dimensional invariant tori ofan associated infinite dimensional dynamical system.

An abstract infinite dimensional KAM theorem with application to nonlinear higher dimensional Schrodinger equation systems