Coupling of Two Partial Differential Equations and its Application, II : the Case of Briot-Bouquet Type PDEs
Coupling of Two Partial Differential Equations and its Application, II : the Case of Briot-Bouquet Type PDEs
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DOI：
10.2977/prims/1241553125
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发表时间：
2009-06
影响因子：
1.2
通讯作者：
H. Tahara
中科院分区：
文献类型：
--
作者：
H. Tahara
设F（t，x，u，v）是在C4的原点的邻域中满足F（0，x，0，0）0和（F/v）（0，x，0，0）<$0;则方程（A）t u/t = F（t，x，u，u/x）称为Briot-Bouquet型偏微分方程，函数λ（x）=（F/u）（0，x，0，0）称为特征指数.本文考虑在假设λ（0）-（−∞，0]<${ 1，2，. }.通过考虑两个方程（A）和（B）的耦合，并通过求解它们的耦合方程来进行简化。将所得结果应用于求（A）的所有奇异解的问题。
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) ≡ 0a nd (∂F/∂v)(0 ,x ,0, 0) ≡ 0; then the equation (A) t∂u/∂t = F (t, x, u, ∂u/∂x) is called a Briot-Bouquet type partial differential equation, and the function λ(x )=( ∂F/∂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∂w/∂t = λ(x)w under the assumption λ(0) � (−∞, 0] ∪{ 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).