On Systems of Partial Differential Equations of Briot–bouquet Type

Abstract

We study systems of partial differential equations of Briot–Bouquet type. The existence of holomorphic solutions to such systems largely depends on the eigenvalues of an associated matrix. For the noninteger case, we generalise the well-known result of Gérard and Tahara [‘Holomorphic and singular solutions of nonlinear singular first order partial differential equations’, Publ. Res. Inst. Math. Sci.26 (1990), 979–1000] for Briot–Bouquet type equations to Briot–Bouquet type systems. For the integer case, we introduce a sequence of blow-up like changes of variables and give necessary and sufficient conditions for the existence of holomorphic solutions. We also give some examples to illustrate our results.

Uniqueness of the solution of nonlinear singular first order partial differential equations

“…In the case where F (t, x, z 1 , z 2 ) is a holomorphic function in a neighborhood of C t × C x × C z 1 × C z 2 , this equation is called a Briot-Bouquet type partial differential equation in t, and it was investigated in details by Gérard-Tahara [3,4] and Yamazawa [16]. See also Li [8] and Yamazawa [17].…”