Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of PI

Costin, Ovidiu; Huang, Min; Tanveer, S.

doi:10.1215/00127094-2429589

Cited by 43 publications

(88 citation statements)

References 14 publications

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“…k on the right-hand side are the low order terms of the second non-hydrodynamic series f (2) (w) in (36). Note that in (47) all constants on the right-hand-side are known: the overall normalization constant is fixed to be twice the very same Stokes constant S 1 found in the largeorder behavior (39) of the hydrodynamic series.…”

Section: Large Order Behavior Stokes Constants and Borel Analysismentioning

confidence: 99%

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Hydrodynamics, resurgence, and transasymptotics

2015

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The second-order hydrodynamical description of a homogeneous conformal plasma that undergoes a boostinvariant expansion is given by a single nonlinear ordinary differential equation, whose resurgent asymptotic properties we study, developing further the recent work of Heller and Spalinski [Phys. Rev. Lett. 115, 072501 (2015)]. Resurgence clearly identifies the non-hydrodynamic modes that are exponentially suppressed at late times, analogous to the quasi-normal-modes in gravitational language, organizing these modes in terms of a trans-series expansion. These modes are analogs of instantons in semi-classical expansions, where the damping rate plays the role of the instanton action. We show that this system displays the generic features of resurgence, with explicit quantitative relations between the fluctuations about different orders of these non-hydrodynamic modes. The imaginary part of the trans-series parameter is identified with the Stokes constant, and the real part with the freedom associated with initial conditions.

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Section: Large Order Behavior Stokes Constants and Borel Analysismentioning

confidence: 99%

“…(19), (34), (36) and (37), with C λ = 2C η . This procedure can be continued to determine all the F k (ζ), as all the equations are linear for k ≥ 1.…”

Section: Trans-asymptotic Rearrangement and Initial Conditionsmentioning

confidence: 99%

Hydrodynamics, resurgence, and transasymptotics

2015

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“…It was proved recently in [13] with a technique developed in [12]; see also [25][26][27] for partial results.…”

Section: Conjecture 11 If the 2-or 3-truncated Solution Of A Painlevmentioning

confidence: 99%

“…In this paper, we will further improve the technique in [13] (see also a recent work [1] for other improvements of [13]) and give an analytic proof of Conjecture 1.1 in the context of a special 2-truncated solution of PII, namely, the Hastings-McLeod solution [22]. This solution might be the most famous one among the Painlevé transcendents, due to its frequent appearances in applications, especially in mathematical physics.…”

Section: Conjecture 11 If the 2-or 3-truncated Solution Of A Painlevmentioning

confidence: 99%

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Location of Poles for the Hastings–McLeod Solution to the Second Painlevé Equation

Huang

Zhang

2015

Constr Approx

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We show that the well-known Hastings-McLeod solution to the second Painlevé equation is pole-free in the region arg x ∈ [− π 3 , π 3 ] ∪ [ 2π 3 , 4π 3 ], which proves an important special case of a general conjecture concerning pole distributions of Painlevé transcedents proposed by Novokshenov. Our strategy is to construct explicit quasi-solutions approximating the Hastings-McLeod solution in different regions of the complex plane and estimate the errors rigorously. The main idea is very similar to the one used to prove Dubrovin's conjecture for the first Painlevé equation, but there are various technical improvements.

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Universality Near the Gradient Catastrophe Point in the Semiclassical Sine‐Gordon Equation

Miller

2021

Comm Pure Appl Math

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We study the semiclassical limit of the sine-Gordon (sG) equation with below threshold pure impulse initial data of Klaus-Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava, and Klein, we found that in a O. 4=5 / neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painlevé I tritronquée solution. A linear map can be explicitly made from the tritronquée solution to this neighborhood. Under this map: away from the tritronquée poles, the first correction of sG is universally given by the real part of the Hamiltonian of the tritronquée solution; localized defects appear at locations mapped from the poles of the tritronquée solution; the defects are proved universally to be a two-parameter family of special localized solutions on a periodic background for the sG equation. We are able to characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann-Hilbert problems, substantially generalizing [5] to establish universality beyond the context of solutions of a single equation. © 2021 Wiley Periodicals LLC. B.-Y. LU AND P. D. MILLER 3.2. Expression for g and determination of 3.3. The modulated librational wave region 3.4. Proof of Proposition 1.3 3.4.1. Asymptotically trivial jump conditions for O.w/ 3.4.2. Construction of a parametrix for O.w/ 3.4.3. Error analysis 3.5. The boundary of the modulated librational wave region and points of gradient catastrophe 3.6. Symmetries for x h 0 4. Proof of Theorem 1.6 4.1. Simplification of jump conditions near w h 4.2. Modified g-function near the gradient catastrophe 4.2.1. A more singular ansatz for g 4.2.2. Redetermination of .x; t/ and £ .x; t/ via construction of a conformal mapping W U 3 C 4.2.3. Values of the partial derivatives of s.x; t/ at the gradient catastrophe 4.3. The phase-linearized outer parametrix 4.4. Use of the modified g-function and inner parametrix for O.w/ near w h 4.5. Characterization of T.s / 4.5.1. Riemann-Hilbert Problem 4.1 and the Painlevé-I equation 4.5.2. Identification of the real tritronquée solution 4.5.3. Laurent expansions and additional identities 4.6. Global parametrix for O.w/, error analysis, and expressions for the potentials 4.6.1. Global parametrix definition 4.6.2. Error analysis 4.6.3. Asymptotic formulae for cos. 1 2 u N .x; t// and sin. 1 2 u N .x; t// 5. Proof of Theorem 1.10 5.1. Modifying the inner parametrix 5.2. Initial global parametrix, corresponding error matrix, and parametrix for the error 5.3. Corrected error matrix, small-norm problem, and formulae for the potentials 5.4. Differential equations satisfied by the leading terms 5.4.1. Relating C and S to sine-Gordon 5.4.2. E.w/ as a Darboux transformation matrix for K.ws X; T /

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Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of PI

Cited by 43 publications

References 14 publications

Hydrodynamics, resurgence, and transasymptotics

Hydrodynamics, resurgence, and transasymptotics

Location of Poles for the Hastings–McLeod Solution to the Second Painlevé Equation

Universality Near the Gradient Catastrophe Point in the Semiclassical Sine‐Gordon Equation

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