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Integrable Partial Differential Equations: Geometry, Asymptotics, and Numerics.

Deliverables

Public Lecture 1

By this date at least a public lecture on subjects related with the project and aimed at the wider public will be delivered at each node.

Conference at Unimib

First general Conference of the Project. The main focus will be on WP2( (Near-to) Integrable PDEs and Applications), state of the art review and novel developments. The state of the art concerning WP3 and WP4 will be also reviewed. The conference will be organized by Unimib. The entire network will participate, and attendance by experts outside the consortium will also be encouraged.

Conference at UBFC

Second general Conference of the project. The main focus will be on WP4 (Numerical Approaches), state of the art review and novel developments. The state of the art in WP2 and WP3 will also reviewed. The conference will be organized by UBFC (in particular University of Bourgogne). The entire network will participate, and attendance by experts outside the consortium will also be encouraged.

Web Page

Web page of the network, created and constantly updated with the description of the goals and the actions of the network, both scientific and organizational. The web page will also have a section devoted to the large public, for outreach purpose.

video gallery

Realization of an online video gallery with typical solutions of dispersive PDEs. This gallery will combine information in the form of wikipedia articles with videos illustrating solutions described in the text. The items of the gallery will be presented during the planned public lectures.

Table-top tank for water waves

Realization of a table-top size tank equipped with surface and possibly internal waves generators. This will make possible to perform small scale experiments aimed at popularizing our research.

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Publications

Numerical Approach to Painlevé Transcendents on Unbounded Domains

Author(s): Christian Klein, Nikola Stoilov
Published in: Symmetry, Integrability and Geometry: Methods and Applications, 2018, ISSN 1815-0659
Publisher: Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine
DOI: 10.3842/sigma.2018.068

Drinfeld-Sokolov Hierarchies, Tau Functions, and Generalized Schur Polynomials

Author(s): Mattia Cafasso, Ann du Crest de Villeneuve, Di Yang
Published in: Symmetry, Integrability and Geometry: Methods and Applications, 2018, ISSN 1815-0659
Publisher: Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine
DOI: 10.3842/sigma.2018.104

The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type

Author(s): Mattia Cafasso, Manuel D. de la Iglesia
Published in: Symmetry, Integrability and Geometry: Methods and Applications, 2018, ISSN 1815-0659
Publisher: Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine
DOI: 10.3842/sigma.2018.076

Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane

Author(s): Marco Bertola, José Gustavo Elias Rebelo, Tamara Grava
Published in: Symmetry, Integrability and Geometry: Methods and Applications, 2018, ISSN 1815-0659
Publisher: Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine
DOI: 10.3842/sigma.2018.091

Noncommutative Painlevé Equations and Systems of Calogero Type

Author(s): M. Bertola, M. Cafasso, V. Rubtsov
Published in: Communications in Mathematical Physics, 363/2, 2018, Page(s) 503-530, ISSN 0010-3616
Publisher: Springer Verlag
DOI: 10.1007/s00220-018-3210-0

Hydrodynamic Models and Confinement Effects by Horizontal Boundaries

Author(s): R. Camassa, G. Falqui, G. Ortenzi, M. Pedroni, C. Thomson
Published in: Journal of Nonlinear Science, 29/4, 2019, Page(s) 1445-1498, ISSN 0938-8974
Publisher: Springer Verlag
DOI: 10.1007/s00332-018-9522-6

Fredholm Determinant Solutions of the Painlevé II Hierarchy and Gap Probabilities of Determinantal Point Processes

Author(s): Mattia Cafasso, Tom Claeys, Manuela Girotti
Published in: International Mathematics Research Notices, 2019, ISSN 1073-7928
Publisher: Oxford University Press
DOI: 10.1093/imrn/rnz168

A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions

Author(s): Estelle Basor, Pavel Bleher, Robert Buckingham, Tamara Grava, Alexander Its, Elizabeth Its, Jonathan P Keating
Published in: Nonlinearity, 32/10, 2019, Page(s) 4033-4078, ISSN 0951-7715
Publisher: Institute of Physics Publishing
DOI: 10.1088/1361-6544/ab28c7

Spectral approach to the scattering map for the semi-classical defocusing Davey–Stewartson II equation

Author(s): Christian Klein, Ken McLaughlin, Nikola Stoilov
Published in: Physica D: Nonlinear Phenomena, 400, 2019, Page(s) 132126, ISSN 0167-2789
Publisher: Elsevier BV
DOI: 10.1016/j.physd.2019.05.006

Quantum D 4 Drinfeld–Sokolov hierarchy and quantum singularity theory

Author(s): Ann du Crest de Villeneuve, Paolo Rossi
Published in: Journal of Geometry and Physics, 141, 2019, Page(s) 29-44, ISSN 0393-0440
Publisher: Elsevier BV
DOI: 10.1016/j.geomphys.2019.02.011

Singularity formation as a wetting mechanism in a dispersionless water wave model

Author(s): R Camassa, G Falqui, G Ortenzi, M Pedroni, G Pitton
Published in: Nonlinearity, 32/10, 2019, Page(s) 4079-4116, ISSN 0951-7715
Publisher: Institute of Physics Publishing
DOI: 10.1088/1361-6544/ab2a1a

Multidomain spectral method for the Gauss hypergeometric function

Author(s): S. Crespo, M. Fasondini, C. Klein, N. Stoilov, C. Vallée
Published in: Numerical Algorithms, 2019, ISSN 1017-1398
Publisher: Baltzer Science Publishers B.V.
DOI: 10.1007/s11075-019-00741-7

On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations

Author(s): Roberto Camassa, Gregorio Falqui, Giovanni Ortenzi, Marco Pedroni
Published in: Symmetry, Integrability and Geometry: Methods and Applications, 2019, ISSN 1815-0659
Publisher: Department of Applied Research, Institute of Mathematics of National Academy of Science of Ukraine
DOI: 10.3842/sigma.2019.087

Brezin–Gross–Witten tau function and isomonodromic deformations

Author(s): Marco Bertola, Giulio Ruzza
Published in: Communications in Number Theory and Physics, 13/4, 2019, Page(s) 827-883, ISSN 1931-4523
Publisher: International Press of Boston, Inc.
DOI: 10.4310/cntp.2019.v13.n4.a4

The τ -function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant

Author(s): Harini Desiraju
Published in: Journal of Mathematical Physics, 60/11, 2019, Page(s) 113505, ISSN 0022-2488
Publisher: American Institute of Physics
DOI: 10.1063/1.5120357

On a class of derivative Nonlinear Schrödinger-type equations in two spatial dimensions

Author(s): Jack Arbunich, Christian Klein, Christof Sparber
Published in: ESAIM: Mathematical Modelling and Numerical Analysis, 53/5, 2019, Page(s) 1477-1505, ISSN 0764-583X
Publisher: EDP Sciences
DOI: 10.1051/m2an/2019018

Numerical scattering for the defocusing Davey–Stewartson II equation for initial data with compact support

Author(s): Christian Klein, Nikola Stoilov
Published in: Nonlinearity, 32/11, 2019, Page(s) 4258-4280, ISSN 0951-7715
Publisher: Institute of Physics Publishing
DOI: 10.1088/1361-6544/ab28c6

A Study of the Direct Spectral Transform for the Defocusing Davey‐Stewartson II Equation the Semiclassical Limit

Author(s): O. Assainova, C. Klein, K. D. T.‐R. McLaughlin, P. D. Miller
Published in: Communications on Pure and Applied Mathematics, 72/7, 2019, Page(s) 1474-1547, ISSN 0010-3640
Publisher: John Wiley & Sons Inc.
DOI: 10.1002/cpa.21822

On the “Vacuum” Dam-Break Problem: Exact Solutions and Their Long Time Asymptotics

Author(s): Roberto Camassa, Gregorio Falqui, Giovanni Ortenzi, Marco Pedroni, Giuseppe Pitton
Published in: SIAM Journal on Applied Mathematics, 80/1, 2020, Page(s) 44-70, ISSN 0036-1399
Publisher: Society for Industrial and Applied Mathematics
DOI: 10.1137/19m1266836

Borodin–Okounkov formula, string equation and topological solutions of Drinfeld–Sokolov hierarchies

Author(s): Mattia Cafasso, Chao-Zhong Wu
Published in: Letters in Mathematical Physics, 109/12, 2019, Page(s) 2681-2722, ISSN 0377-9017
Publisher: Kluwer Academic Publishers
DOI: 10.1007/s11005-019-01205-8

Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds

Author(s): P. Lorenzoni, R. Vitolo
Published in: Journal of Geometry and Physics, 149, 2020, Page(s) 103573, ISSN 0393-0440
Publisher: Elsevier BV
DOI: 10.1016/j.geomphys.2019.103573

Three computational approaches to weakly nonlocal Poisson brackets

Author(s): Matteo Casati, Paolo Lorenzoni, Raffaele Vitolo
Published in: Studies in Applied Mathematics, 2020, ISSN 0022-2526
Publisher: Blackwell Publishing Inc.
DOI: 10.1111/sapm.12302

Tau Functions as Widom Constants

Author(s): M. Cafasso, P. Gavrylenko, O. Lisovyy
Published in: Communications in Mathematical Physics, 365/2, 2019, Page(s) 741-772, ISSN 0010-3616
Publisher: Springer Verlag
DOI: 10.1007/s00220-018-3230-9