Abstract. In this talk we'll discuss the microlocal regularity of $\mathcal{C}^2$ solutions of the full nonlinear equation
\begin{equation*}
u_t = f(x,t,u,u_x),
\end{equation*}
where $f(x,t,\zeta_0, \zeta)$ is $\mathcal{C}^\mathcal{M}$ (Denjoy-Carleman) in the variables $(x,t)\in \mathbb{R}^{N} \times \mathbb{R}$ and holomorphic in the variables $(\zeta_0,\zeta) \in \mathbb{C} \times \mathbb{C}^{N}$. More precisely we show that if $\mathcal{C}^\mathcal{M}$ is a regular Denjoy-Carleman class (including the quasianalytic case) then:
\begin{equation*}
\mathrm{WF}_\mathcal{M} (u)\subset \mathrm{Char}(L^u),
\end{equation*}
where $\mathrm{WF}_\mathcal{M}(u)$ is the Denjoy-Carleman wave-front set of $u$ and $\mathrm{Char}(L^u)$ is the characteristic set of the linearized operator $L^u$:
\begin{equation*}
L^u = \dfrac{\partial}{\partial t} - \sum_{j=1}^{N}\frac{\partial f}{\partial\zeta_j}(x,t,u,u_x)\dfrac{\partial}{\partial x_j}.
\end{equation*}
To do so, we study the existence of approximate solutions of vector fields in $\mathbb{R}^{N+1}\times\mathbb{C}^M$ of the form
\begin{equation*}
    L=\frac{\partial}{\partial t} + \sum_{k=1}^Na_k(x,t,\zeta)\frac{\partial}{\partial x_j} + \sum_{j=1}^Mb_j(x,t,\zeta)\frac{\partial}{\partial \zeta_j},
\end{equation*}
where the coefficients $a_k$ and $b_j$ are holomorphic with respect to the variable $\zeta$ and $\mathcal{C}^\mathcal{M}$  with respect to $(x,t)$. This is joint work with Antonio Victor da Silva Jr.