Tokyo Journal of Mathematics

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

Hidetoshi TAHARA

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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

Permanent link to this document
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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