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We study the local and global well-posedness for the coupled system ofSchr\"odinger and Kawahara equations on the real line. The Sobolev space $L^{2}\times H^{-2}$ is the space where the lowest regularity local solutions areobtained. The energy space is $H^1 \times H^2$. We apply theColliander-Holmer-Tzirakis method [7] to prove the global well-posedness in$L^2 \times L^2$ where the energy is not finite. Our method generalizes themethod of Colliander-Holmer-Tzirakis in the sense that the operator thatdecouples the system is nonlinear.

Global well-posedness for the coupled system of Schr\"odinger and Kawahara equations