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Singular solutions for a class of traveling wave equations arising in hydrodynamics
Geyer, Anna (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Mañosa, Víctor (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.
Note: Agraïments: The first author is supported by the FWF project J3452 "Dynamical Systems Methods in Hydrodynamics" of the Austrian Science Fund.
Note: Número d'acord de subvenció MINECO/DPI2011-25822
Note: Número d'acord de subvenció AGAUR/2014/SGR-859
Rights: Tots els drets reservats.
Language: Anglès
Document: article ; recerca ; acceptedVersion
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

The record appears in these collections:
Research literature > UAB research groups literature > Research Centres and Groups (scientific output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles

 Record created 2017-01-23, last modified 2020-11-01



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