On the formation of singularities of solutions of nonlinear differential systems in antistokes directions

Costin, Ovidiu; Costin, Rodica D.

doi:10.1007/s002220100153

Cited by 78 publications

(210 citation statements)

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“…To apply this analysis in our example, which does not satisfy all the conditions in [14], small adaptations of the proofs are needed (see Appendix). One can determine that for large η, uniformly in the sector arg η ∈ − 4π 9 − δ, 4π 9 − δ for some δ ∈ 0, 2 9 π , except for an exponentially small region around singularity of G 0 , the asymptotic series of G 0 (η) is of the form…”

Section: /9mentioning

confidence: 99%

“…We first mention briefly a number of results in [14] and then allow for slight modifications in the assumptions, to adjust for the equation of G 0 .…”

Section: Appendix: Singularities Of Nonlinear Odesmentioning

confidence: 99%

“…We adopt, with few exceptions that we mention, the same conditions, notations and terminology as [15] and [14]; the results on formal solutions and their generalized Borel summability are also taken from [15].…”

Section: Setting Of [14] and Generalizationsmentioning

confidence: 99%

“…The only difference relevant to [14] with respect to the analysis in [15] is in the study of Y, and once the analog results are obtained, the analysis of Y k is virtually identical. By the normalization choice, we have The system is of the form (131)…”

Section: Normal Form Of Eq (7)mentioning

confidence: 99%

“…Insofar as analysis of the leading order equation (in step (iii) above) is concerned, which (typically a nonlinear ODE), formal calculations have been used before (see [3], and references in [14]). These can now be rigorously derived from the general theory introduced in [14].…”

Section: Introductionmentioning

confidence: 99%

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Complex Singularity Analysis for a Nonlinear PDE

Costin

Tanveer

2006

Communications in Partial Differential Equations

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We introduce a method of rigorous analysis of the location and type of complex singularities for nonlinear higher order PDEs as a function of the initial data. The method is applied to determine rigorously the asymptotic structure of singularities of the modified Harry-Dym equationfor small time at the boundaries of the sector of analyticity. Previous work [7], [5] shows existence, uniqueness and Borel summability of solutions of general PDEs. It is shown that the solution to the above initial value problem is represented convergently by a series in a fractional power of t down to a small annular neighborhood of a singularity of the leading order equation. We deduce that the exact solution has a singularity nearby having, to leading order, the same type.

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Section: /9mentioning

confidence: 99%

“…We first mention briefly a number of results in [14] and then allow for slight modifications in the assumptions, to adjust for the equation of G 0 .…”

Section: Appendix: Singularities Of Nonlinear Odesmentioning

confidence: 99%

Section: Setting Of [14] and Generalizationsmentioning

confidence: 99%

Section: Normal Form Of Eq (7)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Complex Singularity Analysis for a Nonlinear PDE

Costin

Tanveer

2006

Communications in Partial Differential Equations

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Analytic methods for obstruction to integrability in discrete dynamical systems

Costin

Kruskal

2004

Comm Pure Appl Math

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A unique analytic continuation result is proved for solutions of a relatively general class of difference equations, using techniques of generalized Borel summability.This continuation allows for Painlevé property methods to be extended to difference equations.It is shown that the Painlevé property (PP) induces, under relatively general assumptions, a dichotomy within first order difference equations: all equations with PP can be solved in closed form; on the contrary, absence of PP implies, under some further assumptions, that the local conserved quantities are strictly local in the sense that they develop singularity barriers on the boundary of some compact set.The technique produces analytic formulas to describe fractal sets originating in polynomial iterations.

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Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings

Mariño

2014

Fortschritte der Physik

237

293

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In these lectures I present a review of non-perturbative instanton effects in quantum theories, with a focus on large N gauge theories and matrix models. I first consider the structure of these effects in the case of ordinary differential equations, which provide a model for more complicated theories, and I introduce in a pedagogical way some technology from resurgent analysis, like trans-series and the resurgent version of the Stokes phenomenon. After reviewing instanton effects in quantum mechanics and quantum field theory, I address general aspects of large N instantons, and then present a detailed review of non-perturbative effects in matrix models. Finally, I consider two applications of these techniques in string theory

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On the formation of singularities of solutions of nonlinear differential systems in antistokes directions

Abstract: Inventiones mathematicae manuscript No.

Cited by 78 publications

References 9 publications

Complex Singularity Analysis for a Nonlinear PDE

Complex Singularity Analysis for a Nonlinear PDE

Analytic methods for obstruction to integrability in discrete dynamical systems

Lectures on non‐perturbative effects in large N gauge theories, matrix models and strings

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