Roberto Camassa, Gregorio Falqui, Giovanni Ortenzi, Marco Pedroni, “On the Geometry of Extended Self-Similar Solutions of the Airy Shallow Water Equations”, SIGMA, 2019, Volume 15,087, 17 pp.

This article is cited in 4 papers

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

Roberto Camassa^a, Gregorio Falqui^b, Giovanni Ortenzi^b, Marco Pedroni^c

^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

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.

MSC: 37K05; 37J15; 76M55

Received: July 17, 2019; in final form October 31, 2019; Published online November 9, 2019

Language: English

DOI: 10.3842/SIGMA.2019.087