The Modified Camassa–Holm Equation

The Camassa–Holm (CH) equation describes pseudo-spherical surfaces and therefore its integrability properties can be studied by geometrical means [Reyes, “Geometric integrability of the Camassa–Holm equation.” Letters in Mathematical Physics 59 (2002): 117–31]. Using this fact, we introduce a “Miura transform” and a “modified” Camassa–Holm (mCH) equation, in analogy with the Korteweg–de Vries theory. We obtain an infinite number of local conservation laws for CH from these data and also an infinite number of (non)local symmetries. We then compute conservation laws for mCH and also show that it describes pseudo-spherical surfaces, so that, in particular, it is the integrability condition of an $sl(2,\ \mathbb R)$-valued linear problem. Finally, we investigate mCH analytically: we define weak solutions and prove their existence and uniqueness.