It is well known that certain integrable equations such as the Camassa-Holm, Novikov and Qiao admit the so-called peakon solutions. However, all of them are not evolution equations. Then I would like to know if there is some evolution equation admitting such property.
The answer might be negative. Let me explain. An integrable peakon equation usually has the following structure (you can refer to some papers written by Fokas/Fuchssteiner/Olver around 1996):
u_{t}=P \cdot (\partial_x-\partial_{x}^{3})^{-1} u_x
where P is a Hamiltonian operator (also known as implectic operator). The term (\partial_x-\partial_{x}^{3})^{-1} u_x is crucial for admitting peakon, therefore the operator (\partial_x-\partial_{x}^{3})^{-1} should be remained (can not be absorbed into P). Then one has to introduce u=(1- \partial_x^{2})m so as to get rid of nonlocal terms. Thus, the equation is non-evolution in terms of m.
There might be exceptional, but I do not know.
The answer can be positive. Indeed, Camassa-Holm equation can be considered as a (local) evolution equation by taking inverse of the operator 1-\partial_x^2. Note that (1-\partial_x^2)^{-1}=1+\partial_x^2+\partial_x^4+\dots. But, of course, as an evolution equation, Camassa-Holm contains infinitely many terms (each term is a differential polynomial in u).
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