Gevrey Regularity and Local Well-Posedness for the HirotaSatsuma System

In this paper, we examine the local well-posedness of the initial value problem for the HirotaSatsuma system within Gevrey spaces. This system, which consists of a coupled nonlinear dispersive partial differential equation, models the interactions between long and short waves and is known for its integrable structure. We demonstrate that the problem is locally well-posed in the Gevrey spaces G_η,δ,k(ℝ) × G_η,δ,k+1 (ℝ) for k>-18 k > -\frac{1}{8} , and η ≥ 1. This finding improves upon existing well-posedness results in Sobolev spaces H^k(ℝ) × H^k+1(ℝ). Our approach involves a meticulous analysis of linear and bilinear estimates within Gevrey classes. By utilizing Fourier analytical techniques and reformulating the system into an integral equation through Duhamels principle, we establish the necessary bounds for applying a fixed-point argument. This process yields results regarding existence, uniqueness, and continuous dependence on initial data. Furthermore, we show that the solution demonstrates Gevrey−3η regularity in time, capturing the smoothing properties of the system. These results deepen our understanding of analytic-type regularity in nonlinear dispersive systems.