On long time behavior of solutions of the Schrödinger-KdV system with and without resonant interactions

We consider the long time behavior of the solutions of the coupled Schrödinger-KdV systems \begin{eqnarray*} \left{ \begin{array}{llll}i\partial_tu+\partial^2_xu=αuv+βu|u|^2,\hskip30pt (x,t)\in $\mathbb{R}$\times $\mathbb{R}$^{+},\ \partial_tv+\partial^3_xv+v\partial_xv=γ\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.