Open Problems for Painlevé Equations

Clarkson, Peter A.

doi:10.3842/sigma.2019.006

Cited by 14 publications

(16 citation statements)

References 145 publications

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“…We refer the reader to [6] and [8] for asymptotic relations for these solutions as well as for the results concerning the absence of poles in particular sectors of the complex plane. See also [9] for the survey article describing the present knowledge and open questions related to the solutions of the PII equation.…”

Section: Introductionmentioning

confidence: 99%

Total integrals of Ablowitz‐Segur solutions for the inhomogeneous Painlevé II equation

Kokocki

2020

Stud Appl Math

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In this paper, we establish a formula determining the value of the Cauchy principal value integrals of the real and purely imaginary Ablowitz-Segur solutions for the inhomogeneous second Painlevé equation. Our approach relies on the analysis of the corresponding Riemann-Hilbert problem and the construction of an appropriate parametrix in a neighborhood of the origin. Obtained integral formulas are consistent with already known analogous results for the Ablowitz-Segur solutions of the homogeneous Painlevé II equation. K E Y W O R D S asymptotic expansion, geometric flow, Painlevé II equation, Riemann-Hilbert-problem 1 − 2 + 3 + 1 2 3 = −2 sin( ).(2)Roughly speaking, any choice of ( 1 , 2 , 3 ) ∈ ℂ 3 satisfying the condition (2) gives us a solution Φ( , ) of the corresponding RH problem, which is a 2 × 2 matrix-valued function sectionally holomorphic in and meromorphic with respect to the variable (see Refs. 1-5). If we assume that ( , ) ∶= (4 3 ∕3 + ) is a phase function and 3 is the third Pauli matrix (see definition (15)), then the function 504

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Section: Introductionmentioning

confidence: 99%

Total integrals of Ablowitz‐Segur solutions for the inhomogeneous Painlevé II equation

Kokocki

2020

Stud Appl Math

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“…Notice, that the systems ( 21) and ( 22) with scalar variables are equivalent since the relation pq = β(y) is the first integral. In the non-Abelian case, this is only a partial first integral, so the system ( 21) is more general than (22). Another difference is that the scalar system (22) easily reduces to the sinh-Gordon equation in rational form…”

Section: Reduction Of the Toda Latticementioning

confidence: 99%

Non-Abelian Toda lattice and analogs of Painlevé III equation

Adler¹

2022

Preprint

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In integrable models, stationary equations for higher symmetries serve as one of the main sources of reductions consistent with dynamics. We apply this method to the non-Abelian two-dimensional Toda lattice. It is shown that already the stationary equation of the simplest higher flow gives a non-trivial non-autonomous constraint that reduces the Toda lattice to a non-Abelian analog of the pumped Maxwell-Bloch equations. The Toda lattice itself is interpreted as an auto-Bäcklund transformation acting on the solutions of this system. Further self-similar reduction leads to non-Abelian analogs of the Painlevé III equation.

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“…However, unlike the linear case, Painlevé transcendents are more complicated and their connection problems have been studied and solved in some interesting, albeit limited, cases. Especially for the P I transcendents, the connection problems are widely open; P. A. Clarkson announced finding the connection formulas for P I as open problems on several occasions [7,8,9]. Among these problems, we are particular interested in the following two connection problems.…”

Section: Connection Problems Of P Imentioning

confidence: 99%

“…The solutions of the Painlevé equations are also called Painlevé transcendents, and they are considered to be "the nonlinear special functions" that generalize the arsenal of classical special functions (Airy, Bessel, parabolic cylinder, hypergeometric functions, etc. ), see [7,8,9].…”

Section: Introductionmentioning

confidence: 99%

Connection problem of the first Painlevé transcendent between poles and negative infinity

Long¹,

Li²,

Wang³

2021

Preprint

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We consider a connection problem of the first Painlevé equation (P I ), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable t tends to negative infinity for real P I functions. We get a classification of the real P I functions in terms of (p, H) so that they behave differently at the negative infinity, where p is the location of a pole and H is the free parameter in the Laurent series. Some limiting-form connection formulas of P I functions are obtained for large H. Specifically, for the real tritronquée solution, the largen asymptotic formulas of p n and H n are obtained, where p n is the n-th pole on the real line in the ascending order and H n is the associated free parameter. Our approach is based on the complex WKB method (also known as the method of uniform asymptotics) introduced by Bassom, Clarkson, Law and McLeod in their study on the connection problem of the second Painlevé transcendent [Arch. Rational Mech. Anal., 1998, pp. 241-271]. Several numerical simulations are carried out to verify our main results. Meanwhile, we obtain the phase diagram of P I solutions in the (p, H) plane, which somewhat resembles the Brillouin zones in solid-state physics. The asymptotic and numerical results obtained in this paper partially answer Clarkson's open question on the connection problem of the first Painlevé transcendent.

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Open Problems for Painlevé Equations

Cited by 14 publications

References 145 publications

Total integrals of Ablowitz‐Segur solutions for the inhomogeneous Painlevé II equation

Total integrals of Ablowitz‐Segur solutions for the inhomogeneous Painlevé II equation

Non-Abelian Toda lattice and analogs of Painlevé III equation

Connection problem of the first Painlevé transcendent between poles and negative infinity

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