On Totally Characteristic Type Non-linear Partial Differential Equations in the Complex Domain

Abstract

The paper deals with a singular non-linear partial differential equation $t\partial u/\partial t=F(t,x,u,\partial u/\partial x)$ with two independent variables $(t,x)\in {\mathbf{C}}^2$ under the assumption that $F(t,x,u,v)$ is holomorphic and $F(0,x,0,0)=0$. Set $\gamma (x)=(\partial F/\partial v)(0,x,0,0)$. In case $\gamma (x)=0$ the equation was investigated quite well by Gérard–Tahara [3]. In case $\gamma (0)=0$ and $\mathrm{Re}{\gamma }^{\prime }<0$ the existence of holomorphic solution was proved in Chen–Tahara [2] under a non-resonance condition. The present paper proves the existence of holomorphic solution under the same non-resonance condition but using the following weaker condition: $\gamma (0)=0$ and ${\gamma }^{\prime }(0)\in \mathbf{C}\backslash [0,\infty )$. The result is extended to higher order equations.