Methods for the computation of the multivalued Painlevé transcendents on their Riemann surfaces

Abstract

We extend the numerical pole field solver (B. Fornberg and J.A.C. Weideman, J. Comput. Phys. 230:5957-5973, 2011) to enable the computation of the multivalued Painlevé transcendents, which are the solutions to the third, fifth and sixth Painlevé equations, on multiple sheets of their Riemann surfaces. We display, for the first time we believe, solutions to these equations on multiple Riemann sheets. We also provide numerical evidence for the existence of solutions to the sixth Painlevé equation that have pole-free sectors, known as tronquée solutions.

Open Problems for Painlevé Equations

“…Some recent numerical computations of Painlevé equations include: a pole field solver using Padé approximations [65,66,69,70,71,126,127]; numerical Riemann-Hilbert problems [119,120,122,121,133,142]; Fredholm determinants [26,27]; Padé approximations [110,112,144]; pole elimination [5,6,7,8,9,10,11]; a multidomain spectral method [96].…”

A computational exploration of the McCoy–Tracy–Wu solutions of the third Painlevé equation

“…If the large-z behavior of the solution is given by (2), then σ and B are no longer arbitrary but they become functions of λ and ν. Specifically, as demonstrated in [25], the connection formulae relating the large-z behavior, (2), and the small-z behavior, (9), are given by…”

“…Hence, the PFS enabled the study of unexplored P I , P II and P IV solutions, as reported in [11][12][13][32][33][34]. In [9], we extended the PFS method to the computation of multivalued solutions of the P III , P V and P VI equations, thus making the solution spaces of these equations amenable to numerical exploration. This paper is the first application of this enhanced PFS method to the survey of a class of multivalued Painlevé transcendents.…”

“…Solutions that contain a pole-free sector, such as the MTW solutions, have been known as tronquée solutions since 1913 [4], and have been identified and studied for the P I -P V equations, see [2,7,[17][18][19][20]22,35]. Furthermore, numerical evidence in [9] suggests that tronquée solutions of P VI also exist. In particular, for P III with γ = 1 = −δ, Lin, Dai and Tibboel proved in [22] that for every α and β there exist (i) a one-parameter family of tronquée solutions with u ∼ 1, z → ∞, −π/2 < arg z < π/2 and (ii) a unique tronquée solution with u ∼ 1, z → ∞, −π/2 < arg z < 3π/2.…”

Numerical Approach to Painlevé Transcendents on Unbounded Domains

“…In this paper we are interested in the tritronquée solutions in the regular sector and will construct them on unbounded lines via polynomial interpolation. If poles are to be studied, approaches based on Padé approximants as in [11,13,26] are to be used.…”