Local well-posedness for a system of modified KdV equations in modulation spaces

Abstract

In this work, we consider the initial value problem (IVP) for a system of modified Korteweg-de Vries (mKdV) equations \begin{equation} \begin{cases} \partial_t v + \partial_x^3 v+ \partial_x (v w^2) = 0, \hspace{0.98 cm} v(x,0)=ψ(x),\\ \partial_t w + α\partial_x^3 w+\partial_x (v^2 w) = 0,\hspace{0.5 cm} w(x,0)=ϕ(x). \end{cases} \end{equation} The main interest is in addressing the well-posedness issues of the IVP when the initial data are considered in the modulation space $M_s^{2,p}(\mathbb{R})$, $p\ge 2$. In the case when $0<\alpha \ne 1$, we derive new trilinear estimates in these spaces and prove that the IVP is locally well-posed for data in $M_s^{2,p}(\mathbb{R})$ whenever $s>\frac{1}{4}-\frac{1}{p}$ and $p\ge 2$. In deriving the trilinear estimate, the fact that the Fourier supports of the solution components $v$ and $w$ lie on distinct cubic curves, namely $\tau ={\xi }^3$ and $\tau =\alpha {\xi }^3$, introduces additional difficulties in handling the resonant case. This makes the analysis substantially different from what one encounters in the single-equation setting. To overcome the difficulties arising in the resonant case, it was necessary to impose the more restrictive condition $s>\frac{1}{4}-\frac{1}{p}$ on the trilinear estimate, rather than the natural threshold $s>\frac{1}{4}-\frac{3}{2p}$ , which would otherwise yield sharp local well-posedness for $s>-\frac{1}{2}$ when $p=2$.

29 pages