A new integrable hierarchy, parametric solution and traveling wave solution

This paper gives a new 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 . Finally, we give the traveling wave solution (TWS) of the above three equations. The TWSs of the first two equations have singularity, but the TWS of the 3rd one is continuous. For the 5th-order equation, its smooth parametric solution can not include its singular TWS. We also analyse the initial Gaussian solutions for the equations ut = ∂ 5 xu − 2 3 , and uxxt + 3uxxux + uxxxu = 0. The former is stable, but the latter is not.