Singular solutions for a class of traveling wave equations arising in hydrodynamics

We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form üu+12u̇2+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 th...

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Vydáno v:	Nonlinear analysis: real world applications Ročník 31; s. 57 - 76
Hlavní autoři:	Geyer, Anna, Mañosa, Víctor
Médium:	Journal Article Publikace
Jazyk:	angličtina
Vydáno:	Elsevier Ltd 01.10.2016
Témata:	35 Partial differential equations 35Q Equations of mathematical physics and other areas of application 37 Dynamical systems and ergodic theory 37C Smooth dynamical systems: general theory 37N Applications 76 Fluid mechanics 76B Incompressible inviscid fluids Camassa-Holm equations Camassa–Holm equation Classificació AMS Differential equations Differential equations, Partial Equacions diferencials i integrals Equacions diferencials parcials Equacions diferencials singulars Fluid dynamics Fluid flow Hidrodinàmica Hydrodinamics Hydrodynamics Integrable vector fields Matemàtiques i estadística Mathematical analysis Periodic solutions Singular ordinary differential equations Stems Surface waves Traveling waves Àrees temàtiques de la UPC Camassa–Holm equation Traveling waves Integrable vector fields Singular ordinary differential equations
ISSN:	1468-1218, 1878-5719
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Shrnutí:	We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form üu+12u̇2+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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23
ISSN:	1468-1218 1878-5719
DOI:	10.1016/j.nonrwa.2016.01.009