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

Long, Wen-Gao; Zeng, Zhao-Yun

doi:10.1007/s00365-022-09571-8

Cited by 2 publications

(4 citation statements)

References 21 publications

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“…4 − a 3 , where M 1 and M 2 are two constants with 0 < M 2 < M 1 . In case (i), the result generalizes the main theorems in [13], which is similar to the results for the PII equation given in [22]. In case (ii), we get some new properties of the PI solutions.…”

Section: Connection Problem Of Pisupporting

confidence: 80%

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Connection problem of the first Painlevé transcendents with large initial data

Long¹,

Li²

2023

Preprint

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In previous work, Bender and Komijani (2015 J. Phys. A: Math. Theor. 48 475202) showed that the separatrix solutions to the first Painlevé (PI) equations are determined by a PT -symmetric Hamiltonian, and a sequence of initial conditions give rise to separatrix solutions. In the present work, we consider the initial value problem of the PI equation in a more general setting, and show that the initial conditions y(0) and y (0) located on a sequence of curves Γ n , n = 1, 2, . . . , will give rise to separatrix solutions, and we find the limiting form equation of the curves Γ n as n → ∞. These curves separate the singular solutions and oscillating solutions of PI. Our analytical asymptotic formula of Γ n match the numerical results remarkably well, even for small n.

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Section: Connection Problem Of Pisupporting

confidence: 80%

“…A similar study to the second Painlevé equation is carried out in [22]. They give an asymptotic classification of the PII solutions with large initial data (a, b) by assuming that b 2 − a 4 → ±∞.…”

Section: Discussionmentioning

confidence: 99%

Connection problem of the first Painlevé transcendents with large initial data

Long¹,

Li²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…4 − a 3 , where M 1 and M 2 are two constants with 0 < M 2 < M 1 . In case (i), the result generalizes the main theorems in [13], which is similar to the results for the PII equation given in [23]. In case (ii), we get some new properties of the PI solutions.…”

Section: Introductionsupporting

confidence: 80%

“…A similar study to the second Painlevé equation is carried out in [23]. They give an asymptotic classification of the PII solutions with large initial data (a, b) by assuming that b 2 − a 4 → ±∞.…”

Section: Discussionmentioning

confidence: 99%

Connection problem of the first Painlevé transcendents with large initial data

Long

2023

J. Phys. A: Math. Theor.

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In previous work, Bender and Komijani (2015, J. Phys. A: Math. Theor. 48, 475202) studied the first Painlevé (PI) equation and showed that the sequence of initial conditions giving rise to separatrix solutions could be asymptotically determined using a $\mathcal{PT}$-symmetric Hamiltonian. In the present work, we consider the initial value problem of the PI equation in a more general setting. We show that the initial conditions $(y(0),y'(0))=(a,b)$ located on a sequence of curves $\Gamma_n$, $n=1,2,\dots$, will give rise to separatrix solutions. These curves separate the singular and the oscillating solutions of PI. The limiting-form equation $b^2/4 - a^3=f_n \sim A n^{6/5}$ for the curves $\Gamma_{n}$ as $n\to\infty$ is derived, where $A$ is a positive constant. The discrete set $\{f_n\}$ could be regarded as the nonlinear eigenvalues. Our analytical asymptotic formula of $\Gamma_n$ matches the numerical results remarkably well, even for small $n$. The main tool is the method of uniform asymptotics introduced by Bassom et al. (1998, Arch. Rational Mech. Anal. 143, 241--271) in the studies of the second Painlevé equation.

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

Cited by 2 publications

References 21 publications

Connection problem of the first Painlevé transcendents with large initial data

Connection problem of the first Painlevé transcendents with large initial data

Connection problem of the first Painlevé transcendents with large initial data

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