On Borel summation and Stokes phenomena for rank-1 nonlinear systems of ordinary differential equations

Complex Singularity Analysis for a Nonlinear PDE

“…However, unlike the isomonodromic method, the method based on generalized Borel summation summation [13], [15] applies to initially small solutions of non-integrable equations as well. 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).…”

“…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].…”

“…As in [15] we normalize the system so that ℜ(α j ) > 0. Performing a further transformation of the type y → y − M k=1 a k x −k (which takes out M terms of the formal asymptotic series solutions of the equation), makes…”

“…the terms C k e −λ·kx x α·k x −r (with k ∈ (N ∪ {0}) n , r ∈ N ∪ {0}) form a well ordered set with respect to ≫ on d (see also [15]). …”

“…The operator LB constructed in [15] can be applied to any transseries solution (119) of (117) (valid on its open sector S trans , assumed non-empty) on any direction d ⊂ S trans and yields an actual solution y = LBỹ of (117), analytic in a domain S an . Conversely, any solution y(x) satisfying (122) on a direction d is represented as LBỹ(x), on d, for some uniqueỹ(x):…”