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

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

“…In this section, we will recall the formal theory of the coupling of two partial differential equations. For details, see [14] and [15]. Let F (t, x, u, v) and G(t, x, u, v) be holomorphic functions in a neighborhood of (0, 0, 0, 0…”

“…Baouendi-Goulaouic [2] first treated this type of equations in a little bit special form; then Gérard-Tahara [10] treated it in the general case and determined all the singular solutions belonging to the class S + in the case λ(0) ∈ N * , and Yamazawa [16] solved the case λ(0) ∈ N * . Later, Tahara [15] determined all the singular solutions belonging to the class S zero by using the following reduction theorem for the equation (1.1): THEOREM 1.1 ([15]). If A 1 ), A 2 ), A 3 ) and λ(0) ∈ (−∞, 0] ∪ {1, 2, .…”

“…The paper is organized as follows. In the next section 2, we will recall the formal theory of coupling of two partial differential equations developed in [14] and [15]. In section 3 we will look for formal solutions of the coupling equations, in section 4 we will give estimates of these formal solutions, and in section 5 we will complete the proof of Theorem 1.2.…”

“…

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, . .

…”

On Systems of Partial Differential Equations of Briot–bouquet Type

“…, f m ) is holomorphic satisfying F(0, 0) = 0. Since the work of Briot and Bouquet [1], many authors have worked on Briot-Bouquet systems (see, for example, [2,3,8,10,11] and [6] for an extensive bibliography).…”