Let $L(z,{\partial }_z)=({\partial }_{z_0}{)}^k-A(z,{\partial }_z)$ be a linear partial differential operator with holomorphic coefficients, where$$A(z,{\partial }_z)=\sum _{j=0}^{k-1}{A}_j(z,{\partial }_{z^{\text{'}}}\left)\right({\partial }_{z_0}{)}^j,\mathrm{ord}.A(z,{\partial }_z)=m\>k$$and$$z=(z_0,z^{\text{'}})\in C^{n+1}.$$We consider Cauchy problem with holomorphic data$$L(z,{\partial }_z)u\left(z\right)=f\left(z\right),\left({\partial }_{z_0}{)}^{i}u\right(0,z^{\text{'}})={\widehat{u}}_i(z^{\text{'}}\left)\right(0\le i\le k-1).$$We can easily get a formal solution $\widehat{u}\left(z\right)={\sum }_{n=0}^{\infty }{\widehat{u}}_n(z^{\text{'}}\left)\right(z_0{)}^n/n!$, bu in general it diverges. We show under some conditions that for any sector $S$ with the opening less that a constant determined by $L(z,{\partial }_z)$, there is a function $u_S\left(z\right)$ holomorphic except on $\{z_0=0\}$ such that $L(z,{\partial }_z)u_S\left(z\right)=f\left(z\right)$ and $u_S\left(z\right)\sim \widehat{u}\left(z\right)$ as $z_0\to 0$ in $S$.

We study the convergence or divergence of formal (power series) solutions of first order nonlinear partial differential equations    (SE) f(x,u,Dx u) = 0 with u(0)=0. Here the function f(x,u,ξ) is defined and holomorphic in a neighbourhood of a point $(0,0,{\xi }^0)\in {ℂ}_x^n\times {ℂ}_u\times {ℂ}_{\xi }^n({\xi }^0=D_{x}u\left(0\right))$ and $f(0,0,{\xi }^0)=0$. The equation (SE) is said to be singular if f(0,0,ξ) ≡ 0 $\left(\xi \in {ℂ}^n\right)$. The criterion of convergence of a formal solution $u\left(x\right)={\sum }_{\left|\alpha \right|\ge 1}{u}_{\alpha }{x}^{\alpha }$ of (SE) is given by a generalized form of the Poincaré condition which depends on each formal solution. In the case where the formal...

We study convergence of formal power series along families of formal or analytic vector fields. One of our results says that if a formal power series converges along a family of vector fields, then it also converges along their commutators. Using this theorem and a result of T. Morimoto, we prove analyticity of formal solutions for a class of nonlinear singular PDEs. In the proofs we use results from control theory.