On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE’s
On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE’s
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发表时间：
1998
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通讯作者：
O. Costin
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作者：
O. Costin
本文研究了非共振条件下的解析（线性或）非线性常微分方程组在秩为一的非正则奇点处的解。证明了形式渐近指数级数解（transseries解：形式幂级数乘以小指数的可数线性组合）在广义上沿着任何指数衰减方向都是Borel可和的.相反地，任何沿沿着某个方向递减的解都是跨级数的波莱尔和。介绍的求和过程是一个扩展的Borel求和，这是线性的，乘法的，可与微分和复共轭。求和算法只使用形式解（而不是它们所解的微分方程）。沿着沿着奇异（斯托克斯）方向，求和重建的函数被证明是由拉普拉斯积分沿着特殊路径，一个子集的Écalle的中值路径。一对一的对应关系之间建立的实际解决方案和广义博雷尔和transseries是常数之间的斯托克斯线和变化，如果斯托克斯线交叉（局部斯托克斯现象）。我们分析了本地和经典斯托克斯现象之间的联系。研究了transseries的transseries中包含的级数的Borel（形式逆拉普拉斯）变换的解析性质，并给出了它们的奇异性的系统描述。这些Borel变换满足一个层次的卷积方程，我们给出了在超函数空间的一般解决方案。此外，我们还证明了它们是Écalle意义下的复活函数。求和过程是不唯一的，我们分类所有适当的扩展Borel求和的transseries解决方案的非共振系统。我们发现公式连接不同的系列包含在transseries之间（复苏方程）。复苏原来是密切相关的当地斯托克斯现象。
In this paper we study analytic (linear or) nonlinear systems of ordinary differential equations, at an irregular singularity of rank one, under nonresonance conditions. It is shown that the formal asymptotic exponential series solutions (transseries solutions: countable linear combinations of formal power series multiplied by small exponentials) are Borel summable in a generalized sense along any direction in which the exponentials decay. Conversely, any solution that decreases along some direction is the Borel sum of a transseries. The summation procedure introduced is an extension of Borel summation which is linear, multiplicative, commutes with differentiation and complex conjugation. The summation algorithm uses the formal solutions alone (and not the differential equation that they solve). Along singular (Stokes) directions, the functions reconstructed by summation are shown to be given by Laplace integrals along special paths, a subset of Écalle’s median paths. The one-to-one correspondence established between actual solutions and generalized Borel sums of transseries is constant between Stokes lines and changes if a Stokes line is crossed (local Stokes phenomenon). We analyze the connection between local and classical Stokes phenomena. We study the analytic properties of the Borel (formal inverse Laplace) transform of the series contained in the transseries of the transseries and give a systematic description of their singularities. These Borel transforms satisfy a hierarchy of convolution equations, for which we give the general solution in a space of hyperfunctions. In addition, we show that they are resurgent functions in the sense of Écalle. The summation procedure is not unique; we classify all proper extensions of Borel summation to transseries solutions of nonresonant systems. We find formulas connecting the different series contained in the transseries among themselves (resurgence equations). Resurgence turns out to be closely linked to the local Stokes phenomenon.