In this article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous nonlinear partial differential equations with analytic coefficients at the origin of Cn+1. We systematically examine the cases where the inhomogeneity is s-Gevrey for any s≥0, in order to carefully distinguish the influence of the data (and their degree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative rational number sc fully determined by the Newton polygon of a convenient associated linear partial differential equation: for any s≥sc, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s<sc, the formal solutions are generically sc-Gevrey. In the latter case, we give an explicit example in which the solution is s'-Gevrey for no s'<sc. As a practical illustration, we apply our results to the generalized Burgers-Korteweg-de Vries equation.
Submitted November 6, 2021. Published January 19, 2023.
Math Subject Classifications: 35C10, 35G20, 35Q53.
Key Words: Gevrey order; inhomogeneous partial differential equation; nonlinear partial differential equation; generalized Burgers-KdV equation; Newton polygon; formal power series; divergent power series.
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| Pascal Remy |
Laboratoire de Mathématiques de Versailles
Université de Versailles Saint-Quentin
45 avenue des Etats-Unis
78035 Versailles cedex, France
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