Global well-posedness for the coupled system of Schr\"odinger and Kawahara equations
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We study local and global well-posedness for the coupled system of Schr\"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 are obtained. The energy space is $H^1 \times H^2$. We apply the Colliander--Holmer--Tzirakis method \cite{CHT} to prove the global well-posedness in $L^2 \times L^2$ where the energy is not finite. Our method generalizes the method of Colliander--Holmer--Tzirakis in the sense that the operator that decouples the system is nonlinear.
Abstract
We study local and global well-posedness for the coupled system of Schr\"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 are obtained. The energy space is $H^1 \times H^2$. We apply the Colliander--Holmer--Tzirakis method \cite{CHT} to prove the global well-posedness in $L^2 \times L^2$ where the energy is not finite. Our method generalizes the method of Colliander--Holmer--Tzirakis in the sense that the operator that decouples the system is nonlinear.