Instantaneous analytic smoothing of rough data for the modified and cubic gKdV equations

Abstract

We consider the $k$-generalized Korteweg-de Vries equation \begin{equation*} \partial_{t}v+\partial_{x}^{3}v +\partial_{x}(v^{k+1})=0, \qquad (t,x)\in\mathbb R\times\mathbb R, \qquad k\in\mathbb Z_{+}, \end{equation*} emphasizing the modified case $k=2$ and the cubic case $k=3$. We prove that solutions from low-regularity, possibly singular, data $u_0$ become real analytic in $(t,x)$ for all $t\ne 0$, whenever $u_0$ satisfies a Nelson-type condition \begin{equation*} \sum_{k=0}^{\infty}\frac{α^{k}}{k!}\,\big\|(x\partial_x)^{k}u_0\big\|_{X}<\infty, \end{equation*} for some $\alpha >0$. For the cubic equation, this includes data such as $u_0=x_{+}^{\lambda }$, singular at the origin; for mKdV even discontinuous data yield analytic solutions \emph{e.g} $u_{0}(x)=\sgn(x)e^{-x^{2}}$.For mKdV we work in the sharp well-posedness space $X={\hat{H}}_s^r(\mathbb{R})$, $r\in (1,2]$, $s\ge \frac{1}{2}-\frac{1}{2r}$, with $r=2$ recovering analyticity on $H^s(\mathbb{R})$, $s\ge \frac{1}{4}$, the best mKdV space in the sense of Kato; for the cubic equation we work in $X=H^s(\mathbb{R})$, $s>-\frac{1}{6}$, approaching the critical exponent $s=-\frac{1}{6}$ from above. Analyticity thus holds on the largest known data class for which mKdV is well-posed, and on data approaching the corresponding threshold for the cubic equation, extending a known smoothing effect for KdV ($k=1$) to the modified and cubic nonlinearities and to a broader class of singular profiles, avoiding pseudo-differential calculus via Lorentz-space refinements replacing Bourgain-space localization. The mechanism is dispersive: analyticity is generated by the flow, symmetrically in time, and singular profiles become instantaneously analytic for $t\ne 0$.