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 C 4 satisfying F (0, x, 0, 0) ≡ 0 and (∂F /∂v)(0, x, 0, 0) ≡ 0; then the equation (A) t∂u/∂t = F (t, x, u, ∂u/∂x) is called a partial differential equation of Briot-Bouquet type with respect to t, and the function λ(x) = (∂F /∂u)(0, x, 0, 0) is called the characteristic exponent. In [15], it is proved that if λ(0) ∈ (−∞, 0] ∪ {1, 2, . . .} holds the equation (A) is reduced to the simple form (B 1 ) t∂w/∂t = λ(x)w. The present paper considers the case λ(0) = K ∈ {1, 2, . . .} and proves the following result: if λ(0) = K ∈ {1, 2, . . .} holds the equation (A) is reduced to the form (B 2 ) t∂w/∂t = λ(x)w + γ (x)t K for some holomorphic function γ (x). The reduction is done by considering the coupling of two equations (A) and (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).

On Systems of Partial Differential Equations of Briot–bouquet Type

“…Remark 3.6. In [9], Tahara obtained a similar result for the case m = 1. Our approach is very different to that of [9].…”

“…In [9], Tahara obtained a similar result for the case m = 1. Our approach is very different to that of [9]. The main novelty is the introduction of a series of blowup like transformations which pinpoint the obstacles to the existence of holomorphic solutions of systems of partial differential equations of Briot-Bouquet type.…”