The Modified Camassa-Holm Equation: Wave breaking, Classification of Traveling Waves and Explicit Elliptic Peakons

TL;DR

This paper investigates the modified Camassa-Holm equation, demonstrating wave breaking, classifying all traveling wave solutions including peakons, kinks, cuspons, and composite waves, and deriving explicit elliptic peakon solutions.

Contribution

It provides a comprehensive classification of all traveling wave solutions for the mCH equation, including new explicit elliptic peakon solutions.

Findings

Wave breaking occurs in the mCH equation.

All traveling wave solutions are classified, including peakons, kinks, cuspons, and composite waves.

Explicit elliptic peakon solutions are derived.

Abstract

We show that wave breaking occurs for the modified Camassa-Holm (mCH) equation. Next we classify all traveling wave solutions of the modified Camassa-Holm equation in the weak sense via parametrization of their maxima, minima and wave velocity constants. This equation is shown to admit in addition to more popular solutions like smooth traveling waves and peakons, some not so well-known traveling waves as, for example, kinks, cuspons, composite waves and stumpons. Moreover, explicit peakons in terms of Jacobian elliptic functions are found.