Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues
| dc.contributor.area | Mathematics | en_US |
| dc.contributor.author | Cotti, Giordano | |
| dc.contributor.author | Dubrovin, Boris | |
| dc.contributor.author | Guzzetti, Davide | |
| dc.date.accessioned | 2017-06-23T06:17:29Z | |
| dc.date.available | 2017-06-23T06:17:29Z | |
| dc.date.issued | 2017 | |
| dc.description.abstract | We consider an n×n linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at z=∞, holomorphically depending on parameter t within a polydisc in Cn centred at t=0. The eigenvalues of the leading matrix at z=∞ coalesce along a locus Δ contained in the polydisc, passing through t=0. Namely, z=∞ is a resonant irregular singularity for t∈Δ. We analyse the case when the leading matrix remains diagonalisable at Δ. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as t varies in the polydisc, and their limits for t tending to points of Δ. When the deformation is isomonodromic away from Δ, it is well known that a fundamental matrix solution has singularities at Δ. When the system also has a Fuchsian singularity at z=0, we show under minimal vanishing conditions on the residue matrix at z=0 that isomonodromic deformations can be extended to the whole polydisc, including Δ, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed t=0. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting Δ, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that Δ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed. | en_US |
| dc.identifier.uri | https://openscience.sissa.it/handle/1963/35289 | |
| dc.language.iso | en | en_US |
| dc.relation.firstpage | 1 | en_US |
| dc.relation.lastpage | 85 | en_US |
| dc.subject | Classical Analysis and ODEs | en_US |
| dc.title | Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues | en_US |
| dc.type | Preprint | en_US |