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Symmetry, Integrability and Geometry: Methods and Applications, 2019, Volume 15, 087, 17 pp.
DOI: https://doi.org/10.3842/SIGMA.2019.087 (Mi sigma1523) 		 
This article is cited in 4 scientific papers (total in 4 papers)

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

Roberto Camassaa, Gregorio Falquib, Giovanni Ortenzib, Marco Pedronic

a University of North Carolina at Chapel Hill, Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, Chapel Hill, NC 27599, USA
b Dipartimento di Matematica e Applicazioni, Universitá di Milano-Bicocca, Milano, Italy
c Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Universitá di Bergamo, Dalmine (BG), Italy
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DOI: https://doi.org/10.3842/SIGMA.2019.087
URL: https://www.imath.kiev.ua/~sigma/2019/087/
Abstract: Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schrödinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the recurrence relation from a bi-Hamiltonian structure for the equations. This class of solutions reduces the PDEs to a finite ODE system which admits several conserved quantities, which allow to construct explicit solutions by quadratures and provide the bi-Hamiltonian formulation for the reduced ODEs.
Keywords: bi-Hamiltonian geometry, Poisson reductions, self-similar solutions, shallow water models.
Funding agency Grant number
Istituto Nazionale di Alta Matematica "Francesco Severi"
National Science Foundation RTG DMS-0943851
CMG ARC-1025523
DMS-1009750
DMS-1517879
Office of Naval Research N00014-18-1-2490
DURIP N00014-12-1-0749
European Research Council 778010 IPaDEGAN
This work was supported by the National Science Foundation under grants RTG DMS-0943851, CMG ARC-1025523, DMS-1009750, DMS-1517879, the Office of Naval Research under grants N00014-18-1-2490 and DURIP N00014-12-1-0749. This project has also received fundings under grant H2020-MSCA-RISE-2017 Project No. 778010 IPaDEGAN. All authors gratefully acknowledge the auspices of the GNFM Section of INdAM under which part of this work was carried out.
Received: July 17, 2019; in final form October 31, 2019; Published online November 9, 2019
Bibliographic databases:
Document Type: Article
MSC: 37K05; 37J15; 76M55
Language: English
Citation: Roberto Camassa, Gregorio Falqui, Giovanni Ortenzi, Marco Pedroni, “On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations”, SIGMA, 15 (2019), 087, 17 pp.
Citation in format AMSBIB
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