The second member of the Korteweg-de Vries hierarchy (2KdV) on the Torus is given by
\begin{align}
\begin{cases}
u_t -\partial_x^5 u +\alpha \partial_x (u^3) + \beta \partial_x(\partial_x u)^2 + \gamma \partial_x(u\partial_x^2u) = 0\\
u(x,0) = u_0\in H^s(\mathbb{T}),
\end{cases}
\end{align}
for $(\alpha, \beta, \gamma) = (-10,5,10)$ and $u_0$ real valued. For this choice of coefficients, the equation is known to be completely integrable and wellposed in $L^2(\mathbb{T})$ (Kappeler \& Molnar, 2018). In this talk, we'll provide context and discuss the proof wellposedness for $s>35/64$, unconditional wellposedness for $s> 1$, and nonlinear smoothing of order $\varepsilon < \min(2(s-35/64), 1)$, which states that the nonlinear evolution is, up to a phase rotation of the linear evolution, in $H^{s+\varepsilon}(\mathbb{T})$. In fact, our methods apply to more general coefficients, where the best known prior results only establish wellposedness for $s\geq 3/2$ (Kato, '18).