Coupling of Two Partial Differential Equations and its Application, II — the Case of Briot–Bouquet Type PDEs —

Abstract

Let $F(t,x,u,v)$ be a holomorphic function in a neighborhood of the origin of $C^4$ 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 Briot–Bouquet type partial differential equation, and the function $\lambda (x)=(\partial F/\partial u)(0,x,0,0)$ is called the characteristic exponent. This paper considers a reduction of this equation (A) to a simple form (B) $t\partial w/\partial t=\lambda (x)w$ under the assumption $\lambda (0)\notin (-\infty ,0]\cup \{1,2,...\}$. The reduction is done by considering the coupling of two equations (A) and (B), and by solving their coupling equations. The result is applied to the problem of finding all the singular solutions of (A).