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In this work we study symplectic unitary representations for the Galileigroup. As a consequence a Non-Linear Schr\"odinger equation is derived in phasespace. The formalism is based on the non-commutative structure of thestar-product, and using the group theory approach as a guide a physicallyconsistent theory is constructed in phase space. The state is described by aquasi-probability amplitude that is in association with the Wigner function.With these results, we solve the Gross-Pitaevskii equation in phase space andobtained the Wigner function for the system considered.

Analytical Solution for Gross-Pitaevskii Equation in Phase Space and Wigner Function