Abstract:
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We give an exhaustive characterization of singular weak solutions for ordinary
differential equations of the form $\ddot{u}\,u +
\frac{1}{2}\dot{u}^2 + F'(u) =0$, 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. |