Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 087, 17 pages      arXiv:1907.10920      https://doi.org/10.3842/SIGMA.2019.087

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

Roberto Camassa a, Gregorio Falqui b, Giovanni Ortenzi b and 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, Università di Milano-Bicocca, Milano, Italy
c) Dipartimento di Ingegneria Gestionale, dell'Informazione e della Produzione, Università di Bergamo, Dalmine (BG), Italy

Received July 17, 2019, in final form October 31, 2019; Published online November 09, 2019

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.

Key words: bi-Hamiltonian geometry; Poisson reductions; self-similar solutions; shallow water models.

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