K(m, n) equations with fifth order symmetries and their integrability

For K(m, n) equation ut=Dx3(un)+αDx(um), all non-degenerate (n ≠ 0) cases admitting fifth order symmetries are identified, including K(m1, 1), K(m2,−1/2) and K(m3,−2), where m1=0,1,2,3,m2=−1/2,0,1,3/2 and m3=−2,−1,0,1. For five less studied cases, namely K(0,−2),K(−1,−2),K(−2,−2),K(−1/2,−1/2) and K(3/2,−1/2), bi-Hamiltonian structures are established through their invertible links with some famous integrable equations. Hence, all cases, having fifth order symmetries, of K(m, n) equation are integrable in the bi-Hamiltonian sense. As an interesting observation, their Hamiltonian operators are linearly combinations of Dx, Dx3,uDx+Dxu and DxuDx−1uDx, basic ingredients in the bi-Hamiltonian theory of Korteweg-de Vries and modified Korteweg-de Vries equations.