An integrable hierarchy, parametric solution and traveling wave solution

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This paper gives an integrable hierarchy of nonlinear evolution equations. In this hierarchy there are the following representative equations: & & uₜ=⁵ₓ u-2/3, & & uₜ=⁵ₓ(u-1/3)xx -2(u-1/6)ₓ²u; & & 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=⁵ₓ u-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=⁵ₓ u-2/3, and uxxt+3uxxuₓ+uxxxu=0. One is stable, the other not. Finally, we extend the equation uₜ=⁵ₓ u-2/3 to a large class of equations uₜ=∂ₓˡ u⁻ᵐ/ⁿ, l≥1, n=0, m,n ∈, which still have the singular cusp-like traveling wave solutions.