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

TL;DR

This paper thoroughly characterizes singular weak solutions of a class of ODEs relevant to hydrodynamics, constructing various wave types and analyzing their properties, with applications to well-known equations like Camassa-Holm.

Contribution

It provides a complete classification of singular solutions for a specific ODE class in hydrodynamics, including construction and non-existence results for certain wave types.

Findings

Peaked and compactly supported waves cannot coexist for the same equation.

Explicit construction of peaked, cusped, and decayed waves from the ODE.

Application to Camassa-Holm and surface wave equations demonstrating the theory.

Abstract

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.