On long time behavior of solutions of the Schrödinger-KdV system with and without resonant interactions
Felipe Linares, Dequin Zhou
TL;DR
The paper analyzes the long-time behavior of the Schrödinger-KdV system on the real line, focusing on local energy decay within a time-growing region and distinguishing between resonant ($\beta=0$) and non-resonant ($\beta>0$) interactions. It introduces two weighted virial-type functionals with dual weights and uses mollification of initial data to obtain global solutions, deriving crucial $L^1$-integrability and time-derivative identities that drive the decay analysis. In the resonant/unmixed regime, local decay of derivatives and higher norms is established under a smallness condition, while in the unmixed case with positive signs, decay holds without small data for $p\in(0,2/3)$. The results answer open questions of prior work by Linares and Mendez, and hinge on a combination of weighted energy methods, conserved quantities, and careful control of cross-terms through parameter choices and smallness constraints.
Abstract
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 $(α,β,γ)\in \mathbb{R}^{+}\times \overline{\mathbb{R}^{+}}\times \mathbb{R}^{+}$ 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 $(α,β,γ)\in \mathbb{R}^{+}\times \mathbb{R}^{-}\times \mathbb{R}^{+}$. In this case, we give a \lq\lq smallness" criterion involving the product of the parameter $-β$ 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.