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

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 $t\neq0$, whenever $u_0$ satisfies a Nelson-type condition
\begin{equation*}
\sum_{k=0}^{\infty}\frac{\alpha^{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
$X=\widehat H^{r}_{s}(\mathbb R)$, $r\in(1,2]$, $s\geq\frac12-\frac1{2r}$,
with $r=2$ recovering analyticity on $H^s(\mathbb R)$, $s\geq\frac14$, the
best mKdV space in the sense of Kato; for the cubic equation we work in
$X=H^{s}(\mathbb R)$, $s>-\frac16$, approaching the critical exponent
$s=-\frac16$ 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 $t\neq0$.