Abstract
We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.
Highlights
We have shown that these equations possess families of holomorphic solutions y p (t, z, e) that can be expressed through Laplace transforms of order k and Fourier inverse integrals k y p (t, z, e) =
We focus on equations of the form
The principal objective of this subsection consists of the construction of a unique solution of the equation (29) within the Banach space described in Definition 3
Summary
Q1 (∂z )y(t, z, e) Q2 (∂z )y(t, z, e) + eδD k tδD (k+1) ∂t D R D (∂z )y(t, z, e) δ. In the series of papers [17,18,19,20], the authors classify and provide normal forms for generic unfolding of nonresonant linear systems of ODEs with irregular singularity at the origin. On domains Te × Hβ , provided that e ∈ E pin for a collection of bounded sectors E pin which constitutes a convenient good covering E in , see Theorem 2 In both constructions, the function a( T, e) isproperly selected in a way that the differential operator De,α (∂t ) acts on uout/in as a multiplication by τ on the p out/in.
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