Singular solutions of $q$-difference-differential equations of Briot-Bouquet type (New development of microlocal analysis and singular perturbation theory)

Abstract

In 1990, Gérard - Tahara [3] introduced the Briot - Bouquet type partial differential equation tpartial_{t}u=F(t, x, u, partial_{x}u). In [17] the author showed existences of holomorphic and singular solutions of the following type of difference - differential equations tD_{q}u=F(t, x, u, partial_{x}u) when the characteristic exponent rho(0) neq (q^{N} - 1)/(q- 1) holds. In this paper the author shows existences of singular solutions with rho(0)=(q^{N}-1)/(q-1)