A Coupled Generalized Korteweg-de Vries System Driven by White Noise

TL;DR

This paper analyzes a stochastic, Hamiltonian, coupled generalized Korteweg-de Vries system driven by white noise and proves local well-posedness for initial data in $H^{s}(\mathbb{R})\times H^{s}(\mathbb{R})$ with $s>\tfrac{1}{2}$. The authors formulate a mild Itô form using the Airy propagator and work in Bourgain spaces $X^{s,b}$ and $X^{s,b}_{\alpha}$ to capture dispersive smoothing, deriving sharp linear estimates for the propagators and time-localized Duhamel bounds. They establish comprehensive multilinear estimates for the nonlinearities, including $\partial_x(\phi^{2k+1})$, $\partial_x(\varphi^{2k+1})$, and mixed terms, via Fourier analysis, duality, and region decompositions, which enable a fixed-point argument. Stochastic convolution estimates ensure pathwise regularity and control of the noise term through the Hilbert-Schmidt operator $\Xi\in L_2^{0,s}$. Overall, the work advances stochastic dispersive PDE theory by lowering the regularity threshold to $s>\tfrac{1}{2}$ and extending local well-posedness to a coupled gKdV-type system with white-noise forcing.

Abstract

In this paper, we investigate the Cauchy problem for the coupled generalized Korteweg-de Vries system driven by white noise. We prove local well-posedness for data in $ H^{s} \times H^{s},$ with $ s>1/2$. The key ingredients that we used in this paper are multilinear estimates in Bourgain spaces, the Itô formula and a fixed point argument. Our result improves the local well-posedness result of Gomes and Pastor \cite{gomes2021solitary}.

Theorems & Definitions (12)

• Theorem 1.1
• Proposition 2.1: Linear Estimates ref3
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4: Multilinear estimate
• proof
• Lemma 3.5: Stochastic convolution