## Tokyo Journal of Mathematics

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

Hidetoshi TAHARA

#### 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).

#### Article information

Source
Tokyo J. Math., Volume 36, Number 2 (2013), 539-570.

Dates
First available in Project Euclid: 31 January 2014

https://projecteuclid.org/euclid.tjm/1391177987

Digital Object Identifier
doi:10.3836/tjm/1391177987

Mathematical Reviews number (MathSciNet)
MR3161574

Zentralblatt MATH identifier
1295.35014

Subjects
Primary: 35A20: Analytic methods, singularities

#### Citation

TAHARA, Hidetoshi. On a Reduction of Nonlinear Partial Differential Equations of Briot-Bouquet Type. Tokyo J. Math. 36 (2013), no. 2, 539--570. doi:10.3836/tjm/1391177987. https://projecteuclid.org/euclid.tjm/1391177987

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