Hamiltonian PDEs: deformations, integrability, solutions

Dubrovin, Boris

doi:10.1088/1751-8113/43/43/434002

Cited by 23 publications

(29 citation statements)

References 29 publications

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“…Understanding the global behavior in C of the tritronquée solutions (see below) of the Painlevé equation P I is essential in a number of problems such as the critical behavior in the NLS/Toda lattices ( [6], [7]) and the analysis of the cubic oscillator ( [15]). Considerations related to the behavior of NLS/Toda solutions corroborated by numerical evidence led to the conjecture in the Dubrovin-Grava-Klein paper [6] that the tritronquée solutions are analytic in a neighborhood of the origin O and in a sector of width 8π/5 containing O (cf.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Costin¹,

Huang²,

Tanveer³

2014

Duke Math. J.

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We show that the tritronquée solution yt of the Painlevé equation P I that behaves algebraically for large z with arg z = π/5, is analytic in a region containing the sector z = 0, arg z ∈ − 3π 5 , π and the disk z : |z| < 37 20 . This implies the Dubrovin conjecture, an important open problem in the theory of Painlevé transcendents. The method, building on a technique developed in [4], is general and constructive. As a byproduct, we obtain the value of the tritronquée and its derivative at zero, also important in applications, within less than 1/100 rigorous error bounds. arXiv:1209.1009v2 [math.CA] 23 Oct 2014 24 O. COSTIN 1 , M. HUANG 2 , S. TANVEER 1 8.1. Values of intermediate constants for ρ = 3. The numerical values of these constants might be helpful to the reader who would like to double-check the estimates. These are: J M = 0.282580···, jm = 0.64374···, Y 1,M = 1.16314···, Y 1,R,M = 0.132618···, E M = 0.0490292··· z 2,R,M = 0.54226···, z 2,M = 0.91863···, Mq = 0.066702···, M L,q = 0.075708···, V M = 0.2239···, T M = 0.0385··· M 1 = 1.13838···, M 2 = 0.04303···, M 3 = 0.28346···, M 4 = 0.45227···, M 5 = 0.05430···, M 6 = 0.00231···, M 7 = 0.02018··· 9. Acknowledgments

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Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Costin¹,

Huang²,

Tanveer³

2014

Duke Math. J.

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“…The details of the full proof, which is rather long, will be given elsewhere. be the map that associates to ε ∈ U the unique solution of the Cauchy problem for (9). Moreover, let K = n i=1 N i (2i+1) and N = n i=1 N i .…”

Section: The ε-Expansion Of Kdv Solutionsmentioning

confidence: 99%

“…where a and the initial value ϕ are assumed to be smooth functions, and ϕ is either periodic or rapidly decreasing at infinity. We discuss the aspects of the Dubrovin-Zhang construction which are more related with the present paper; in the next section we will show how the results obtained in Section 3 and 4 provide a rigorous justification to this method for a particular class of equations of type (9). Let us consider equation (28).…”

Section: Hamiltonian Perturbation Of Quasilinear Conservation Lawsmentioning

confidence: 99%

Semiclassical Limit for Generalized KdV Equations Before the Gradient Catastrophe

Masoero

Raimondo

2013

Lett Math Phys

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We study the semiclassical limit of the (generalised) KdV equation, for initial data with Sobolev regularity, before the time of the gradient catastrophe of the limit conservation law. In particular, we show that in the semiclassical limit the solution of the KdV equation: i) converges in H s to the solution of the Hopf equation, provided the initial data belongs to H s , ii) admits an asymptotic expansion in powers of the semiclassical parameter, if the initial data belongs to the Schwartz class. The result is also generalized to KdV equations with higher order linearities.

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“…Our results can be summarised as follows. Theorem 2 A Hamiltonian H 0 = h(v, w) dxdy of type II possesses a nontrivial integrable deformation to the order ǫ 2 if and only if, along with the integrability conditions (8), it satisfies the additional differential constraints…”

Section: Reconstruction Of ǫ 2 -Deformationsmentioning

confidence: 99%

Dispersive deformations of Hamiltonian systems of hydrodynamic type in 2+1 dimensions

Ferapontov¹,

Novikov²,

Stoilov³

2012

Physica D: Nonlinear Phenomena

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We develop a theory of integrable dispersive deformations of 2 + 1 dimensional Hamiltonian systems of hydrodynamic type following the scheme proposed by Dubrovin and his collaborators in 1 + 1 dimensions. Our results show that the multi-dimensional situation is far more rigid, and generic Hamiltonians are not deformable. As an illustration we discuss a particular class of two-component Hamiltonian systems, establishing the triviality of first order deformations and classifying Hamiltonians possessing nontrivial deformations of the second order.MSC: 35L40, 37K05, 37K10, 37K55.

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Hamiltonian PDEs: deformations, integrability, solutions

Cited by 23 publications

References 29 publications

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

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

Semiclassical Limit for Generalized KdV Equations Before the Gradient Catastrophe

Dispersive deformations of Hamiltonian systems of hydrodynamic type in 2+1 dimensions

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