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Generalized Poincaré condition and convergence of formal solutions of some nonlinear totally characteristic equations. (English) Zbl 1373.35081
Summary: This paper discusses a holomorphic nonlinear singular partial differential equation \((t \partial_t)^mu=F(t,x,\{(t \partial_t)^j \partial_x^{\alpha}u \}_{j+\alpha \leq m, j<m})\) that is of nonlinear totally characteristic type. The Newton Polygon at \(x=0\) of the equation is defined, and by means of this polygon we define a generalized Poincaré condition (GP) and a condition (R) that the equation has a regular singularity at \(x=0\). Under these conditions, (GP) and (R), it is proved that every formal power series solution is convergent in a neighborhood of the origin.

MSC:
35C10 Series solutions to PDEs
35A20 Analyticity in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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