Analytic linearization of nonlinear perturbations of Fuchsian systems

Nonlinear perturbation of Fuchsian systems are studied in regions including two singularities. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable). Nevertheless, it is shown that in the case when the linear part has commuting monodromy, and the eigenvalues have positive real parts, there exists a unique correction function of the nonlinear part so that the corrected system becomes analytically linearizable

Singular normal form for the Painlev\'e equation P1

We show that there exists a rational change of coordinates of Painlev\'e's P1 equation $y''=6y^2+x$ and of the elliptic equation $y''=6y^2$ after which these two equations become analytically equivalent in a region in the complex phase space where $y$ and $y'$ are unbounded. The region of equivalence comprises all singularities of solutions of P1 (i.e. outside the region of equivalence, solutions are analytic). The Painlev\'e property of P1 (that the only movable singularities are poles) follows as a corollary. Conversely, we argue that the Painlev\'e property is crucial in reducing P1, in a singular regime, to an equation integrable by quadratures

Uniformization and Constructive Analytic Continuation of Taylor Series

We analyze the general mathematical problem of global reconstruction of a function with least possible errors, based on partial information such as n terms of a Taylor series at a point, possibly also with coefficients of finite precision. We refer to this as the "inverse approximation theory problem, because we seek to reconstruct a function from a given approximation, rather than constructing an approximation for a given function. Within the class of functions analytic on a common Riemann surface Omega, and a common Maclaurin series, we prove an optimality result on their reconstruction at other points on Omega, and provide a method to attain it. The procedure uses the uniformization theorem, and the optimal reconstruction errors depend only on the distance to the origin. We provide explicit uniformization maps for some Riemann surfaces Omega of interest in applications. One such map is the covering of the Borel plane of the tritronquee solutions to the Painleve equations PI-PV. As an application we show that this uniformization map leads to dramatic improvement in the extrapolation of the PI tritronquee solution throughout its domain of analyticity and also into the pole sector. Given further information about the function, such as is available for the ubiquitous class of resurgent functions, significantly better approximations are possible and we construct them. In particular, any one of their singularities can be eliminated by specific linear operators, and the local structure at the chosen singularity can be obtained in fine detail. More generally, for functions of reasonable complexity, based on the nth order truncates alone we propose new efficient tools which are convergent as n to infty, which provide near-optimal approximations of functions globally, as well as in their most interesting regions, near singularities or natural boundaries.Comment: 39 pages, 9 figures; v2 some clarifications adde

Noise Effects on Pade Approximants and Conformal Maps

We analyze the properties of Pade and conformal map approximants for functions with branch points, in the situation where the expansion coefficients are only known with finite precision or are subject to noise. We prove that there is a universal scaling relation between the strength of the noise and the expansion order at which Pade or the conformal map breaks down. We illustrate this behavior with some physically relevant model test functions and with two non-trivial physical examples where the relevant Riemann surface has complicated structureComment: 23 pages, 8 figure

Going to the Other Side via the Resurgent Bridge

Using resurgent analysis we offer a novel mathematical perspective on a curious bijection (duality) that has many potential applications ranging from the theory of vertex algebras to the physics of SCFTs in various dimensions, to q-series invariants in low-dimensional topology that arise, e.g. in Vafa-Witten theory and in non-perturbative completion of complex Chern-Simons theory. In particular, we introduce explicit numerical algorithms that efficiently implement this bijection. This bijection is founded on preservation of relations, a fundamental property of resurgent functions. Using resurgent analysis we find new structures and patterns in complex Chern-Simons theory on closed hyperbolic 3-manifolds obtained by surgeries on hyperbolic twist knots. The Borel plane exhibits several intriguing hints of a new form of integrability. An important role in this analysis is played by the twisted Alexander polynomials and the adjoint Reidemeister torsion, which help us determine the Stokes data. The method of singularity elimination enables extraction of geometric data even for very distant Borel singularities, leading to detailed non-perturbative information from perturbative data. We also introduce a new double-scaling limit to probe 0-surgeries from the limiting $r\to\infty$ behavior of 1/r surgeries, and apply it to the family of hyperbolic twist knots.Comment: 115 pages; 32 figure

On the spectral properties of L_{+-} in three dimensions

This paper is part of the radial asymptotic stability analysis of the ground state soliton for either the cubic nonlinear Schrodinger or Klein-Gordon equations in three dimensions. We demonstrate by a rigorous method that the linearized scalar operators which arise in this setting, traditionally denoted by L_{+-}, satisfy the gap property, at least over the radial functions. This means that the interval (0,1] does not contain any eigenvalues of L_{+-} and that the threshold 1 is neither an eigenvalue nor a resonance. The gap property is required in order to prove scattering to the ground states for solutions starting on the center-stable manifold associated with these states. This paper therefore provides the final installment in the proof of this scattering property for the cubic Klein-Gordon and Schrodinger equations in the radial case, see the recent theory of Nakanishi and the third author, as well as the earlier work of the third author and Beceanu on NLS. The method developed here is quite general, and applicable to other spectral problems which arise in the theory of nonlinear equations