Singular solutions for a class of traveling wave equations arising in hydrodynamics
Geyer, Anna (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Mañosa Fernández, Víctor 1971- (Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada III)

Date:	2016
Abstract:	We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form 12^2 F'(u) =0, where F is an analytic function. Our motivation stems from the fact that in the context of hydrodynamics several prominent equations are reducible to an equation of this form upon passing to a moving frame. We construct peaked and cusped waves, fronts with finite-time decay and compact solitary waves. We prove that one cannot obtain peaked and compactly supported traveling waves for the same equation. In particular, a peaked traveling wave cannot have compact support and vice versa. To exemplify the approach we apply our results to the Camassa-Holm equation and the equation for surface waves of moderate amplitude, and show how the different types of singular solutions can be obtained varying the energy level of the corresponding planar Hamiltonian systems.
Grants:	Ministerio de Economía y Competitividad DPI2011-25822 Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-859
Note:	Agraïments: The first author is supported by the FWF project J3452 "Dynamical Systems Methods in Hydrodynamics" of the Austrian Science Fund.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Camassa-Holm equation ; Integrable vector fields ; Singular ordinary differential equations ; Traveling waves
Published in:	Nonlinear Analysis: Real World Applications, Vol. 31 (2016) , p. 57-76, ISSN 1468-1218

DOI: 10.1016/j.nonrwa.2016.01.009

Postprint 24 p, 563.0 KB

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