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

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.