On the multisummability of WKB solutions of certain singularly perturbed linear ordinary differential equations

Takei, Yoshitsugu

doi:10.7494/opmath.2015.35.5.775

Cited by 6 publications

(13 citation statements)

References 4 publications

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“…We should mention that a similar phenomenon of parametric multilevel Gevrey asymptotics has been observed recently by K. Suzuki and Y. Takei in [12] and Y. Takei in [14] for WKB solutions of the Schrödinger equation…”

Section: Introductionsupporting

confidence: 80%

“…We also describe the behavior of the elements in it under certain operators. In Section 3, we study the formal solution of the auxiliary Cauchy problem (13), (14) with coefficients being elements in the Banach space described in the previous section. After recalling some definitions and properties on the k−Borel-Laplace summability procedure in Section 4.1, we provide the solutions of a singular Cauchy problem (22), (23) which conform the support of the solution for the main problem in our work (32), (33).…”

Section: Introductionmentioning

confidence: 99%

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Some Notes on the Multi-Level Gevrey Solutions of Singularly Perturbed Linear Partial Differential Equations

Lastra

Malek

2017

Trends in Mathematics

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We study the asymptotic behavior of the solutions related to a family of singularly perturbed linear partial differential equations in the complex domain. The analytic solutions obtained by means of a Borel-Laplace summation procedure are represented by a formal power series in the perturbation parameter. Indeed, the geometry of the problem gives rise to a decomposition of the formal and analytic solutions so that a multi-level Gevrey order phenomenon appears. This result leans on a Malgrange-Sibuya theorem in several Gevrey levels.

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Section: Introductionsupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Some Notes on the Multi-Level Gevrey Solutions of Singularly Perturbed Linear Partial Differential Equations

Lastra

Malek

2017

Trends in Mathematics

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“…In this section, we state a version of the classical Ramis-Sibuya theorem (see [13], Theorem XI-2-3) with two different Gevrey levels which describes also the case when multisummability holds on some sector. We mention that a similar multi-level version of the Ramis-Sibuya theorem has already been stated in the manuscript [32] and also in a former work of the authors, see [15].…”

Section: There Existsupporting

confidence: 66%

“…Also from this point of view, only few advances have been performed. Among them, we must mention two recent works by K. Suzuki and Y. Takei, [30] and Y. Takei, [32], for WKB solutions of the Schrödinger equation ǫ 2 ψ ′′ (z) = (z − ǫ 2 z 2 )ψ(z) which possesses 0 as fixed turning point and z ǫ = ǫ −2 as movable turning point tending to infinity as ǫ tends to 0.…”

Section: Introductionmentioning

confidence: 99%

On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems

Lastra¹,

Malek²

2015

Adv Differ Equ

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We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter whose coefficients depend holomorphically on ( , t) near the origin in C 2 and are bounded holomorphic on some horizontal strip in C w.r.t. the space variable. In our previous contribution (Lastra and Malek in Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, arXiv:1403.2350), we assumed the forcing term of the Cauchy problem to be analytic near 0. Presently, we consider a family of forcing terms that are holomorphic on a common sector in time t and on sectors w.r.t. the parameter whose union form a covering of some neighborhood of 0 in C * , which are asked to share a common formal power series asymptotic expansion of some Gevrey order as tends to 0. We construct a family of actual holomorphic solutions to our Cauchy problem defined on the sector in time and on the sectors in mentioned above. These solutions are achieved by means of a version of the so-called accelero-summation method in the time variable and by Fourier inverse transform in space. It appears that these functions share a common formal asymptotic expansion in the perturbation parameter. Furthermore, this formal series expansion can be written as a sum of two formal series with a corresponding decomposition for the actual solutions which possess two different asymptotic Gevrey orders, one stemming from the shape of the equation and the other originating from the forcing terms. The special case of multisummability in is also analyzed thoroughly. The proof leans on a version of the so-called Ramis-Sibuya theorem which entails two distinct Gevrey orders. Finally, we give an application to the study of parametric multi-level Gevrey solutions for some nonlinear initial value Cauchy problems with holomorphic coefficients and forcing term in ( , t) near 0 and bounded holomorphic on a strip in the complex space variable. MSC: 35C10; 35C20

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“…A novel version of Ramis-Sibuya Theorem has been developed in [25], and has provided successful results in previous works by the authors, [14], [15,12]. A version of the result in two different levels which fits our needs is now given without proof, which can be found in [14], [15].…”

Section: K−summable Formal Series and Ramis-sibuya Theoremmentioning

confidence: 87%

On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

Lastra

Malek

2018

Results Math

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We study a family of nonlinear initial value partial differential equations in the complex domain under the action of two asymmetric time variables. Different Gevrey bounds and multisummability results are obtain depending on each element of the family, providing a more complete picture on the asymptotic behavior of the solutions of PDEs in the complex domain in several complex variables.The main results lean on a fixed point argument in certain Banach space in the Borel plane, together with a Borel summability procedure and the action of different Ramis-Sibuya type theorems. variable coefficients (see [5,6,21]). The second author [22], both authors [13] and the two authors and J. Sanz [17] have also contributed in this theory. Recently, multisummability of formal solutions of PDEs has also been put forward in different works. W. Balser [4] described a multisummability phenomenon in certain PDEs with constant coefficients. S. Ouchi [23] constructed multisummable formal solutions of nonlinear PDEs, coming from perturbation of ordinary differential equations. H. Tahara and H. Yamazawa [24] have made progresses on general linear PDEs with non constant coefficients under entire initial data. In [20], G. Lysik constructs summable formal solutions of the one dimensional Burgers equation by means of the Cole-Hopf transform. O. Costin and S. Tanveer [8] construct summable formal power series in time variable to 3D Navier Stokes equations. The authors have obtained results in this direction [14, 15].A recent overview on summability and multisummability techniques under different points of view is displayed in [18].The purpose of the present work is to study the solutions of a family of singularly perturbed partial differential equations from the asymptotic point of view. More precisely, we consider a problem of the formThe elements which conform the nonlinear part P 1 , P 2 are polynomials in their second variable with coefficients being holomorphic functions defined on some neighborhood of the origin, say D(0, 0 ), continuous up to their boundary.Here, D(0, 0 ) stands for the open disc in the complex plane centered at 0, and with positive radius 0 > 0. We write D(0, 0 ) for its closure. Moreover, P stands for some polynomial of six variables, with complex coefficients, and the forcing term f (t 1 , t 2 , z, ) is a holomorphic and bounded function in D(0, ρ) 2 × H β × D(0, 0 ), for some ρ > 0, and where H β stands for the horizontal stripfor some β > 0.The precise configuration of the elements involved in the problem is stated and described in Section 2.2.This paper provides a step beyond in the study of the asymptotic behavior of the solutions of a subfamily of singularly perturbed partial differential equations of the form (1). We first recall some previous advances made in this respect, which motivate the present framework.In [13], we studied under the asymptotic point of view the solutions of certain family of PDEs of the formwhere the elements involved in the problem depend only on one time variable t. Our next aim w...

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On the multisummability of WKB solutions of certain singularly perturbed linear ordinary differential equations

Cited by 6 publications

References 4 publications

Some Notes on the Multi-Level Gevrey Solutions of Singularly Perturbed Linear Partial Differential Equations

Some Notes on the Multi-Level Gevrey Solutions of Singularly Perturbed Linear Partial Differential Equations

On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems

On Parametric Gevrey Asymptotics for Some Initial Value Problems in Two Asymmetric Complex Time Variables

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