Real Solutions of the First Painlevé Equation with Large Initial Data

Long, Wen-Gao; Li, Y.‐T.; Liu, S.‐Y.; Zhao, Yulin

doi:10.1111/sapm.12171

Cited by 9 publications

(34 citation statements)

References 21 publications

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“…Actually, the asymptotic behavior of the Stokes multipliers in Theorem 2.2 is valid not only for fixed C, but also uniformly for all C(ξ) in the corresponding regions, and thus we may assume C(ξ) depends on ξ. Let C(ξ) = 1 2 pξ −4/5 with p fixed, the following corollary is also a direct consequence of Theorem 2.2, which gives an asymptotic classification of P I solutions for fixed p and large negative H. It can be regarded as another kind of nonlinear eigenvalue phenomenon similar to the initial value problem in [2,3,24]. Corollary 2.5.…”

Section: Resultsmentioning

confidence: 91%

“…In particular, Bender and Komijani [2] observed that the three types P I solutions appear alternatively as one initial data fixed and the other varying continuously. Recently, Long et al [24] gave rigorous proof to this phenomenon, obtained an asymptotic classification of the P I solutions with respect to the initial data, and built some limiting-form connection formulas.…”

Section: Connection Problems Of P Imentioning

confidence: 99%

“…Inspired by the ideas in [30] and [24], we find that the essential work is to approximate the Stokes multipliers for large p or H, which is the primary work in the present paper.…”

Section: Connection Problems Of P Imentioning

confidence: 99%

“…Hence, in the second step, we use the known Stokes phenomena of these special functions to calculate the Stokes multipliers of Y , and then calculate those of Φ. A notable difference of the current work from [24] is that the asymptotic formulas of the Stokes multipliers here are valid when one parameter large and the other arbitrary, instead of one large and the other fixed in [24].…”

Section: Uniform Asymptotics and Proof Of The Theoremsmentioning

confidence: 99%

“…To do so, we define two conformal mappings ω(η) by along the two Stokes curves. Then we have the following lemma; see [1] or [24].…”

Section: Case Ii: H → +∞mentioning

confidence: 99%

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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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Section: Resultsmentioning

confidence: 91%

Section: Connection Problems Of P Imentioning

confidence: 99%

“…Inspired by the ideas in [30] and [24], we find that the essential work is to approximate the Stokes multipliers for large p or H, which is the primary work in the present paper.…”

Section: Connection Problems Of P Imentioning

confidence: 99%

Section: Uniform Asymptotics and Proof Of The Theoremsmentioning

confidence: 99%

“…To do so, we define two conformal mappings ω(η) by along the two Stokes curves. Then we have the following lemma; see [1] or [24].…”

Section: Case Ii: H → +∞mentioning

confidence: 99%

See 3 more Smart Citations

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

Long¹,

Li²,

Wang³

2021

Preprint

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On the Connection Problem for the Second Painlevé Equation with Large Initial Data

Long

Zeng

2022

Constr Approx

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Exactly solvable nonlinear eigenvalue problems

Wang

2021

J. Phys.: Conf. Ser.

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The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental equation can be integrated once to a first order nonlinear equation, e.g., the Ricatti equation. It is shown that the nonlinear eigenvalue problems of these semi-transcendental equations are equivalent to linear eigenvalue problems. They share the exactly same eigenvalues. The eigensolutions in the two problems are closely related. The nonlinear eigenvalue problem equivalent to the (half) harmonic oscillator in quantum mechanics is solved exactly. This is the first solvable nonlinear eigenvalue problem. The nonlinear eigenvalue problems of some extended equations are also studied.

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Real Solutions of the First Painlevé Equation with Large Initial Data

Cited by 9 publications

References 21 publications

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

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

On the Connection Problem for the Second Painlevé Equation with Large Initial Data

Exactly solvable nonlinear eigenvalue problems

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