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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Bibliographic Details
Published in:Nonlinear analysis: real world applications Vol. 31; pp. 57 - 76
Main Authors: Geyer, Anna, Mañosa, Víctor
Format: Journal Article Publication
Language:English
Published: Elsevier Ltd 01.10.2016
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:1468-1218
1878-5719
DOI:10.1016/j.nonrwa.2016.01.009