Integrable hierarchy, $3\times 3$ constrained systems, and parametric and peaked stationary solutions

Abstract

This paper gives a new integrable hierarchy of nonlinear evolution equations. The DP equation: $m_t+um_x+3mu_x=0, m=u-u_{xx}$, proposed recently by Desgaperis and Procesi \cite{DP[1999]}, is the first one in the negative hierarchy while the first one in the positive hierarchy is:\ $m_t=4(m^{-{2/3}})_x-5(m^{-{2/3}})_{xxx}+ (m^{-{2/3}})_{xxxxx}$. The whole hierarchy is shown Lax-integrable through solving a key matrix equation. To obtain the parametric solutions for the whole hierarchy, we separatedly discuss the negative and the positive hierarchies. For the negative hierarchy, its $3\times3$ Lax pairs and corresponding adjoint representations are nonlinearized to be Liouville-integrable Hamiltonian canonical systems under the so-called Dirac-Poisson bracket defined on a symplectic submanifold of $\R^{6N} $. Based on the integrability of those finite-dimensional canonical Hamiltonian systems we give the parametric solutions of the all equations in the negative hierarchy. In particular, we obtain the parametric solution of the DP equation. Moreover, for the positive hierarchy, we consider the different constraint and use a similar procudure to the negative case to obtain the parametric solutions of the positive hierarchy. In particular, we give the parametric solution of the 5th-order PDE $m_t=4(m^{-{2/3}})_x-5(m^{-{2/3}})_{xxx}+ (m^{-{2/3}})_{xxxxx}$. Finally, we discuss the stationary solutions of the 5th-order PDE, and particularly give its four peaked stationary solutions. The stationary solutions may be included in the parametric solution, but the peaked stationary solutions not. The 5th-order PDE does not have the cusp soliton although it looks like a higher order Harry-Dym equation.