On long time behavior of solutions of the Schr\"odinger-KdV system with and without resonant interactions

We consider the long time behavior of the solutions of the coupled Schr"odinger-KdV systems \begin{eqnarray*} \left\{ \begin{array}{llll}i\partial_tu+\partial^2_xu=\alpha uv+\beta u|u|^2,\hskip30pt (x,t)\in \mathbb{R}\times \mathbb{R}^{+},\\ \partial_tv+\partial^3_xv+v\partial_xv=\gamma \partial_x(|u|^2), \hskip20pt (x,t)\in \mathbb{R}\times \mathbb{R}^{+},\\ u, v)|_{t=0} =(u_{0}, v_{0}). \end{array} \right. \end{eqnarray*} We show that global solutions to this system satisfy locally energy decay in a suitable interval, growing unbounded in time, in two situations. In the first case, we regard the parameter vector $(\alpha,\beta,\gamma)\in \mathbb{R}^{+}\times \overline{\mathbb{R}^{+}}\times \mathbb{R}^{+}$ without any size assumption on the initial data in $ H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$. In the second one, we consider the parameter vector $(\alpha,\beta,\gamma)\in \mathbb{R}^{+}\times \mathbb{R}^{-}\times \mathbb{R}^{+}$. In this case, we give a \lq\lq smallness" criterion involving the product of the parameter $-\beta$ and a constant depending on the initial data in $H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$. Our results answer positively the open questions raised in [F. Linares, A. J. Mendez, SIAM J. Math. Anal. 53(2021) 3838-3855]. We use new ideas and different techniques from the latter paper.