On a Reduction of Nonlinear Partial Differential Equations of Briot-Bouquet Type

Abstract

Let $F(t,x,u,v)$ be a holomorphic function in a neighborhood of the origin of $\mathbb{C}^4$ satisfying $F(0,x,0,0) \equiv 0$ and $(\partial F/\partial v)(0,x,0,0) \equiv 0$; then the equation (A) $t \partial u/\partial t=F(t,x,u, \partial u/\partial x)$ is called a partial differential equation of Briot-Bouquet type with respect to $t$, and the function $\lambda (x)=(\partial F/\partial u)(0,x,0,0)$ is called the characteristic exponent. In [15], it is proved that if $\lambda(0) \not\in (-\infty,0] \cup \{1,2, \ldots \}$ holds the equation (A) is reduced to the simple form $\mbox{(B}_1)$ $t \partial w/\partial t= \lambda (x)w$. The present paper considers the case $\lambda(0)=K \in \{1,2, \ldots \}$ and proves the following result: if $\lambda(0)=K \in \{1,2, \ldots \}$ holds the equation (A) is reduced to the form $\mbox{(B}_2)$ $t \partial w/\partial t= \lambda (x)w+\gamma(x) t^K$ for some holomorphic function $\gamma(x)$. The reduction is done by considering the coupling of two equations (A) and $\mbox{(B}_2)$, and by solving their coupling equations. The result is applied to the problem of finding all the holomorphic and singular solutions of (A).