arXiv subjects

Alberto Lastra

Publications and source records attributed to Alberto Lastra.

At least 55 records · Page 3Linked to original sources

arXiv2019

On singularly perturbed linear initial value problems with mixed irregular and Fuchsian time singularities

We consider a family of linear singularly perturbed PDE relying on a complex perturbation parameter $ε$. As in a former study of the authors (A. Lastra, S. Malek, Parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems, J. Differential Equations 259 (2015), no. 10, 5220--5270), our problem possesses an irregular singularity in time located at the origin but, in the present work, it entangles also differential operators of Fuchsian type acting on the time variable. As a new feature, a set of sectorial holomorphic solutions are built up through iterated Laplace transforms and Fourier inverse integrals following a classical multisummability procedure introduced by W. Balser. This construction has a direct issue on the Gevrey bounds of their asymptotic expansions w.r.t $ε$ which are shown to bank on the order of the leading term which combines both irregular and Fuchsian types operators.

math.CV↗

arXiv2018

Maillet type theorem for nonlinear totally characteristic partial differential equations

The paper discusses a holomorphic nonlinear singular partial differential equation $(t \partial_t)^mu=F(t,x,\{(t \partial_t)^j \partial_x^αu \}_{j+α\leq m, j<m})$ under the assumption that the equation is of nonlinear totally characteristic type. By using the Newton Polygon at $x=0$, the notion of the irregularity at $x=0$ of the equation is defined. In the case where the irregularity is greater than one, it is proved that every formal power series solution belongs to a suitable formal Gevrey class. The precise bound of the order of the formal Gevrey class is given, and the optimality of this bound is also proved in a generic case.

math.CV↗

arXiv2018

Multisummability in Carleman ultraholomorphic classes by means of nonzero proximate orders

We introduce a general multisummability theory of formal power series in Carleman ultraholomorphic classes. The finitely many levels of summation are determined by pairwise comparable, nonequivalent weight sequences admitting nonzero proximate orders and whose growth indices are distinct. Thus, we extend the powerful multisummability theory for finitely many Gevrey levels, developed by J.-P. Ramis, J. Écalle and W. Balser, among others. We provide both the analytical and cohomological approaches, and obtain a reconstruction formula for the multisum of a multisummable series by means of iterated generalized Laplace-like operators.

math.CV↗

arXiv2018

On parametric Gevrey asymptotics for initial value problems with infinite order irregular singularity and linear fractional transforms

This paper is a continuation a previous work of the authors where parametric Gevrey asymptotics for singularly perturbed nonlinear PDEs has been studied. Here, the partial differential operators are combined with particular Moebius transforms in the time variable. As a result, the leading term of the main problem needs to be regularized by means of a singularly perturbed infinite order formal irregular operator that allows us to construct a set of genuine solutions in the form of a Laplace transform in time and inverse Fourier transform in space. Furthermore, we obtain Gevrey asymptotic expansions for these solutions of some order $K>1$ in the perturbation parameter.

math.CV↗

arXiv2018

On parametric Gevrey asymptotics for some initial value problems in two asymmetric complex time variables

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.

math.CV↗

arXiv2018

On parametric Gevrey asymptotics for some nonlinear initial value problems in two complex time variables

The asymptotic behavior of a family of singularly perturbed PDEs in two time variables in the complex domain is studied. The appearance of a multilevel Gevrey asymptotics phenomenon in the perturbation parameter is observed. We construct a family of analytic sectorial solutions in $ε$ which share a common asymptotic expansion at the origin, in different Gevrey levels. Such orders are produced by the action of the two independent time variables.

math.CV↗

arXiv2018

On parametric Borel summability for linear singularly perturbed Cauchy problems with linear fractional transforms

We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. We construct a collection of holomorphic solutions on a full covering by sectors of a neighborhood of the origin in $\mathbb{C}$ with respect to the perturbation parameter $ε$. This set is built up through classical and special Laplace transforms along piecewise linear paths of functions which possess exponential or super exponential growth/decay on horizontal strips. A fine structure which entails two levels of Gevrey asymptotics of order 1 and so-called order $1^{+}$ is witnessed. Furthermore, unicity properties regarding the $1^{+}$ asymptotic layer are observed and follow from results on summability w.r.t a particular strongly regular sequence recently obtained in a previous study.

math.AP↗

arXiv2017

Multiscale Gevrey asymptotics in boundary layer expansions for some initial value problem with merging turning points

We consider a nonlinear singularly perturbed PDE leaning on a complex perturbation parameter $ε$. The problem possesses an irregular singularity in time at the origin and involves a set of so-called moving turning points merging to 0 with $ε$. We construct outer solutions for time located in complex sectors that are kept away from the origin at a distance equivalent to a positive power of $|ε|$ and we build up a related family of sectorial holomorphic inner solutions for small time inside some boundary layer. We show that both outer and inner solutions have Gevrey asymptotic expansions as $ε$ tends to 0 on appropriate sets of sectors that cover a neighborhood of the origin in $\mathbb{C}^{\ast}$. We observe that their Gevrey orders are distinct in general.

math.CV↗

arXiv2017

Gevrey multiscale expansions of singular solutions of PDEs with cubic nonlinearity

We study a singularly perturbed PDE with cubic nonlinearity depending on a complex perturbation parameter $ε$. This is the continuation of a precedent work by the first author. We construct two families of sectorial meromorphic solutions obtained as a small perturbation in $ε$ of two branches of an algebraic slow curve of the equation in time scale. We show that the nonsingular part of the solutions of each family shares a common formal power series in $ε$ as Gevrey asymptotic expansion which might be different one to each other, in general.

math.AP↗

arXiv2016

On multiscale Gevrey and q-Gevrey asymptotics for some linear q-difference differential initial value Cauchy problems

We study the asymptotic behavior of the solutions related to a singularly perturbed q-difference-differential problem in the complex domain. The analytic solution can be splitted according to the nature of the equation and its geometry so that both, Gevrey and q-Gevrey asymptotic phenomena are observed and can be distinguished, relating the analytic and the formal solution. The proof leans on a two level novel version of Ramis-Sibuya theorem under Gevrey and q-Gevrey orders.

math.AP↗

arXiv2015

On parametric multilevel q-Gevrey asymptotics for some linear Cauchy problem

We study a linear $q-$difference-differential Cauchy problem, under the action of a perturbation parameter $ε$. This work deals with a $q-$analog of the research made in a previoues work, giving rise to a generalization of a recent work by the second author. This generalization is related to the nature of the forcing term which suggests the use of a $q-$analog of an acceleration procedure. The proof leans on a $q-$analog of the so-called Ramis-Sibuya theorem which entails two distinct $q-$Gevrey orders. The work concludes with an application of the main result when the forcing term solves a related problem.

math.CV↗

arXiv2015

Strongly regular multi-level solutions of singularly perturbed linear partial differential equations

We study the asymptotic behavior of the solutions related to a family of singularly perturbed partial differential equations in the complex domain. The analytic solutions are asymptotically represented by a formal power series in the perturbation parameter. The geometry of the problem and the nature of the elements involved in it give rise to different asymptotic levels related to the so-called strongly regular sequences. The result leans on a novel version of a multi-level Ramis-Sibuya theorem.

math.AP↗

arXiv2015

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

We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter $ε$ whose coefficients depend holomorphically on $(ε,t)$ near the origin in $\mathbb{C}^{2}$ and are bounded holomorphic on some horizontal strip in $\mathbb{C}$ w.r.t the space variable. 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 $ε$ whose union form a covering of some neighborhood of 0 in $\mathbb{C}^{\ast}$, which are asked to share a common formal power series asymptotic expansion of some Gevrey order as $ε$ tends to 0. 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 $(ε,t)$ near 0 and bounded holomorphic on a strip in the complex space variable.

math.AP↗

arXiv2014

Multi-level Gevrey solutions of singularly perturbed linear partial differential equations

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.

math.CV↗

arXiv2014

On parametric Gevrey asymptotics for some Cauchy problems in quasiperiodic function spaces

We investigate Gevrey asymptotics for solutions to nonlinear parameter depending Cauchy problems with $2π$-periodic coefficients, for initial data living in a space of quasiperiodic functions. By means of the Borel-Laplace summation procedure, we construct sectorial holomorphic solutions which are shown to share the same formal power series as asymptotic expansion in the perturbation parameter. We observe a small divisor phenomenon which emerges from the quasiperiodic nature of the solutions space and which is the origin of the Gevrey type divergence of this formal series. Our result rests on the classical Ramis-Sibuya theorem which asks to prove that the difference of any two neighboring constructed solutions satisfies some exponential decay. This is done by an asymptotic study of a Dirichlet-like series whose exponents are positive real numbers which accumulate to the origin.

math.AP↗

arXiv2014

On parametric Gevrey asymptotics for some nonlinear initial value Cauchy problems

We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter $ε$ with vanishing initial data at complex time $t=0$ and whose coefficients depend analytically on $(ε,t)$ near the origin in $\mathbb{C}^{2}$ and are bounded holomorphic on some horizontal strip in $\mathbb{C}$ w.r.t the space variable. This problem is assumed to be non-Kowalevskian in time $t$, therefore analytic solutions at $t=0$ cannot be expected in general. Nevertheless, we are able to construct a family of actual holomorphic solutions defined on a common bounded open sector with vertex at 0 in time and on the given strip above in space, when the complex parameter $ε$ belongs to a suitably chosen set of open bounded sectors whose union form a covering of some neighborhood $Ω$ of 0 in $\mathbb{C}^{\ast}$. These solutions are achieved by means of Laplace and Fourier inverse transforms of some common $ε-$depending function on $\mathbb{C} \times \mathbb{R}$, analytic near the origin and with exponential growth on some unbounded sectors with appropriate bisecting directions in the first variable and exponential decay in the second, when the perturbation parameter belongs to $Ω$. Moreover, these solutions satisfy the remarkable property that the difference between any two of them is exponentially flat for some integer order w.r.t $ε$. With the help of the classical Ramis-Sibuya theorem, we obtain the existence of a formal series (generally divergent) in $ε$ which is the common Gevrey asymptotic expansion of the built up actual solutions considered above.

math.AP↗

arXiv2014

Summability in general Carleman ultraholomorphic classes

A definition of summability is put forward in the framework of general Carleman ultraholomorphic classes in sectors, so generalizing $k-$summability theory as developed by J.-P. Ramis. Departing from a strongly regular sequence of positive numbers, we construct an associated analytic proximate order and corresponding kernels, which allow us to consider suitable Laplace and Borel-type transforms, both formal and analytic, whose behavior closely resembles that of the classical ones in the Gevrey case. An application to the study of the summability properties of the formal solutions to some moment-partial differential equations is included.

math.CV↗

arXiv2013

On parametric Gevrey asymptotics for singularly perturbed partial differential equations with delays

We study a family of singularly perturbed $q-$difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter $ε$. Moreover, we achieve the existence of a common formal power series in $ε$ which represents each actual solution, and establish $q-$Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya Theorem regarding $q-$Gevrey asymptotics. A particular Dirichlet like series is studied on the way.

math.AP↗