Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation: I

Abstract

The degenerate third Painlevé equation,and a ∈ C, and the associated tau-function are studied via the Isomonodromy Deformation Method. Connection formulae for asymptotics of the general as τ → ±0 and ±i0 solution and general regular as τ → ±∞ and ±i∞ solution are obtained.

Series Expansions of Painlevé Transcendents near the Point at Infinity

“…There have been numerous papers discussing two-parameter solutions in the studies of connection problems, isomonodromy deformations and others: for example, [3], [37], [20] for (I); [16] for (I), (II); [10], [19], [17], [11] for (II); [25], [22], [21], [24] for (III), (III Ã ), (III 0 ); [23], [4], [12] for (IV); [32], [2] for (V), (V 0 ) (see also [14], [7]). For pole-free asymptotic solutions among them, which oscillate along Stokes curves, our theorems provide their series expansions valid in the curved sectors.…”

Asymptotics of Integrals of Some Functions Related to the Degenerate Third Painlevé Equation

“…Towards this end, one has to explain how to calculate the deviation of the functions ϕ and u along L(τ 0 , τ ). In this paper, the aforementioned problem is considered asymptotically, that is, when the limits of integration belong to small neighbourhoods of the singular points, 0 and ∞, of Equation (1). For this purpose, one requires asymptotics of the functions u and ϕ.…”

“…Asymptotics of the function u were studied in [1,2]: the corresponding asymptotics for the function ϕ can also be extracted from these papers. In order to do so, recall that in Proposition 1.2 of [1] there was one more function:…”

“…The final transformation of the above equation is necessary because it is for the functions √ −AB and B that asymptotic results are given in Proposition 4.3.1, Corollary 4.3.1, and Propositions 5.5 and 5.7 of [1]. It is important to note that in Appendix B of the subsequent paper [2], inconsistencies in the paper [1] were located and rectified.…”

Effect of a small dispersion on self-focusing in a spatially one-dimensional case

“…Из соотношения (30) и чeтности функций r(t, 0), ϕ t (t, 0), можно, воспользовавшись ре-зультатами статьи [30], в явном виде выписать асимптотику ξ(t) для больших значеий −t. А из последней уже вытекает заключение о том, что внутри полукубической параболы (11) на фоне приближения НГО, задаваемых правыми частями (12), происходят двухфазовые малоамплитудные колебания с двумя частотами вида…”