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Confluence of Singularities of Nonlinear Differential Equations via Borel–Laplace Transformations

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Abstract

Borel summable divergent series usually appear when studying solutions of analytic ODE near a multiple singular point. Their sum, uniquely defined in certain sectors of the complex plane, is obtained via the Borel–Laplace transformation. This article shows how to generalize the Borel–Laplace transformation in order to investigate bounded solutions of parameter dependent nonlinear differential systems with two simple (regular) singular points unfolding a double (irregular) singularity. We construct parametric solutions on domains attached to both singularities, that converge locally uniformly to the sectoral Borel sums. Our approach provides a unified treatment for all values of the complex parameter.

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Notes

  1. If instead f(x, 𝜖,0) was only O(|x|+|𝜖|), and \(u_{\pm \sqrt \epsilon }\in \mathbb {C}^{m}\) were the unique solutions of \(0=M u_{\pm \sqrt \epsilon }+f(\pm \sqrt \epsilon ,u_{\pm \sqrt \epsilon },\epsilon )\), with u ±0 = 0, then the change of variable \(y\,\mapsto \, y-\tfrac {1}{2\sqrt \epsilon }\left (u_{+\sqrt \epsilon }(x\,+\,\sqrt \epsilon ) -\!\right .\) \(\left . \! u_{-\sqrt \epsilon }(x\,-\,\sqrt \epsilon )\right )\), analytic in (x, 𝜖), would bring the system (10) to a one with f(x, 𝜖,0)=O(x 2 −𝜖).

  2. For m = 1, it’s been shown in [22, Proposition 3.1], cf. also [10, Lemma 1], that the family (13) is in fact locally orbitally analytically equivalent to a family (10).

  3. These α will later correspond to the direction of the unfolded Laplace integrals (8), and \(\mathbf {T}_{\alpha }^{\pm }(\varLambda , \sqrt \epsilon )\) to their strips of convergence.

  4. More precisely to a covering space of the x-plane ramified at \(\{\sqrt \epsilon ,-\!\sqrt \epsilon \}\), the Riemann surface of t(x, 𝜖)(9).

  5. Zhang also unfolds the Laplace integral (3), unlike us he chooses to unfold the kernel \(e^{-\frac {\xi }{x}}d\xi \) by \(\left (\frac {x-\sqrt \epsilon \xi }{x+\sqrt \epsilon \xi }\right )^{\frac {1}{2 \sqrt \epsilon }}d\xi =e^{-t(x,\xi ^{2}\epsilon )\cdot \xi }d\xi \), in our notation.

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Acknowledgments

I am very grateful to Christiane Rousseau for many helpful discussions and to Reinhard Schäfke and Loïc Teyssier for their interest in my work. The paper was prepared during my doctoral studies at Université de Montreal and finalized during my stay at Université de Strasbourg I want to thank both institutions for their hospitality.

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  1. Université de Strasbourg, Strasbourg, France

    Martin Klimeš

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Klimeš, M. Confluence of Singularities of Nonlinear Differential Equations via Borel–Laplace Transformations. J Dyn Control Syst 22, 285–324 (2016). https://doi.org/10.1007/s10883-015-9290-7

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