Revisiting the Cauchy problem for the Zakharov-Rubenchik/Benney-Roskes system
Hung Luong
TL;DR
This paper addresses the local well-posedness of the 2D/3D Zakharov-Rubenchik/Benney-Roskes system by leveraging dispersive estimates within Bourgain spaces to obtain a contraction mapping in an energy-compatible setting, with the main component $\psi$ in $H^1(\mathbb{R}^d)$. The authors formulate a coupled Schrödinger–wave system via a Klein–Gordon–type decoupling and introduce specialized Bourgain spaces $X_1^{k,b}$ and $X_2^{k,b}$ for the analysis, establishing a fixed-point result in these spaces for $(\psi,\rho,\phi)$ when initial data lie in $H^1(\mathbb{R}^d)\times H^l(\mathbb{R}^d)\times H^{l+1}(\mathbb{R}^d)$ with dimension-dependent $l$. They derive a comprehensive network of nonlinear estimates, showing each nonlinear term gains a positive power of the time cutoff $T$, and determine explicit admissible ranges of $(b_1,b_2,k_2)$ (equivalently $(b_1,b_2,l)$) ensuring contraction for $d=2,3$. The results improve previous regularity thresholds and offer a pathway toward analyzing the full-dispersion Benney-Roskes model and time-scale justification in water-wave contexts.
Abstract
In this paper, we revisit the Cauchy problem for the Zakharov-Rubenchik/Benney-Roskes system. Our method is based on the dispersive estimates and the suitable Bourgain's spaces. We then, obtain the local well-posedness of the solution with the main component $ψ$ belongs to $H^1(\mathbb{R}^d)$ ($d=2, 3$) which is actually the energy space corresponding to this component. Our result also suggests a potential approach to the problem of finding exact existence time scale for the solution of Benney-Roskes model in the context of water waves.