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

Symmetry Integrability and Geometry-Methods and Applications

article

Roberto Camassa¹  Gregorio Falqui²  Giovanni Ortenzi²  Marco Pedroni³ 

[1] University of North Carolina at Chapel Hill, Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics;Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca;Dipartimento di Ingegneria Gestionale, Università di Bergamo

关键词: bi-Hamiltonian geometry;    Poisson reductions;    self-similar solutions;    shallow water models;   

DOI  :  10.3842/SIGMA.2019.087

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown