An integrable hierarchy , parametric solution and traveling wave solution

This paper gives an integrable hierarchy of nonlinear evolution equations. In this hierarchy there are the following representative equations: ut = ∂ 5 xu − 2 3 , ut = ∂ 5 x (u 1 3 )xx − 2(u 1 6 )x u ; uxxt + 3uxxux + uxxxu = 0. The first two are in the positive order hierarchy while the 3rd one is in the negative order hierarchy. The whole hierarchy is shown integrable through solving a key 3×3 matrix equation. The 3×3 Lax pairs and their adjoint representations are nonlinearized to be two Liouville-integrable canonical Hamiltonian systems. Based on the integrability of 6N -dimensional systems we give the parametric solution of the positive hierarchy. In particular, we obtain the parametric solution of the equation ut = ∂ 5 xu − 2 3 . Moreover, we give the traveling wave solution (TWS) of the above three equations. The TWSs of the first two equations have singularity and look like cusp (cusp-like), but the TWS of the 3rd one is continuous. For the 5th-order equation, its parametric solution can not include its singular TWS. We also analyse the Gaussian initial solutions for the equations ut = ∂ 5 xu − 2 3 , and uxxt + 3uxxux + uxxxu = 0. One is stable, the other not. Finally, we extend the equation ut = ∂ 5 xu − 2 3 to a large class of equations ut = ∂ l xu −m/n, l ≥ 1, n 6= 0, m, n ∈ Z, which still have the singular cusp-like traveling wave solutions.