Existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system

In this paper, we undertake a comprehensive study for existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system

\begin{document}$ -\Delta u+\omega u = a|u|^{p}u +E_1(|u|^{2})u\; \; \; in\; \mathbb{R}^2\; or \; \mathbb{R}^3,\;\;\;\;\;\;{\rm{(DS)}} $\end{document}

which appears in the description of the evolution of surface water waves. In the case of \begin{document}$ L^2 $\end{document}-critical case, i.e., \begin{document}$ N = 2 $\end{document}, \begin{document}$ a>0 $\end{document} and \begin{document}$ 0<p<2 $\end{document}, we show that normalized ground states blow up as \begin{document}$ c \nearrow c^*: = \|R\|^2_{L^2} $\end{document}, where \begin{document}$ R $\end{document} is the ground state solution to equation (DS) with \begin{document}$ a = 0 $\end{document}. We then give a detailed description for the asymptotic behavior of normalized ground states as \begin{document}$ c \nearrow c^* $\end{document}. In the case of \begin{document}$ L^2 $\end{document}-supercritical case, i.e., \begin{document}$ N = 3 $\end{document}, we prove several existence and stability/instability results. We also give new criteria about global existence and blow-up for the associated evolutional equation.