Asymptotics of ramified type for some nonlinear Cauchy problem with Mahler transforms

We fix attention on a nonlinear partial differential Cauchy problem with polynomial coefficients in complex time and space paired with time acting Mahler transforms. A bounded holomorphic solution to the problem is achieved near the origin through a convergent power series in the space variable with Taylor coefficients being Laplace transforms in time. Each Taylor coefficient is shown to carry a formal Puiseux series as asymptotic expansion in time that comprise a finite number of ramifications. The resulting exponential series of these Puiseux expansions in the space variable turn out to formally solve the given Cauchy problem and represent generically a so-called Hahn series relatively to time with rational support.