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

Abstract

We consider the long time behavior of the solutions of the coupled Schrödinger-KdV system$\{\begin{array}{c}i{\partial }_{t}u+{\partial }_x^{2}u=\alpha uv+\beta u|u{|}^2,(x,t)\in R\times R^{+},\\ {\partial }_{t}v+{\partial }_x^{3}v+v{\partial }_{x}v=\gamma {\partial }_x\left(\right|u{|}^2),(x,t)\in R\times R^{+},\\ (u,v){|}_{t=0}=(u_0,v_0).\end{array}$ 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 $(\alpha ,\beta ,\gamma )\in R^{+}\times \overline{R^{+}}\times R^{+}$ without any size assumption on the initial data in $H^1\left(R\right)\times H^1\left(R\right)$. In the second one, we consider the parameter vector $(\alpha ,\beta ,\gamma )\in R^{+}\times R^{-}\times R^{+}$. In this case, we give a ‘‘smallness” criterion involving the product of the parameter −β and a constant depending on the initial data in $H^1\left(R\right)\times H^1\left(R\right)$. Our results answer positively the open questions raised in F. Linares, A. J. Mendez (2021) [18]. We use new ideas and different techniques from the latter paper.