An integrable hierarchy, parametric solution and traveling wave solution

Abstract

This paper gives an integrable hierarchy of nonlinear evolution equations. In this hierarchy there are the following representative equations: \beqq & & u_t=\pa^5_x u^{-{2/3}}, & & u_t=\pa^5_x\frac{(u^{-{1/3}})_{xx} -2(u^{-{1/6}})_{x}^2}{u}; & & u_{xxt}+3u_{xx}u_x+u_{xxx}u=0. \eeqq 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\times 3 $ matrix equation. The $3\times3$ 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 $u_t=\pa^5_x u^{-{2/3}}$. 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 $u_t=\pa^5_x u^{-{2/3}}$, and $u_{xxt}+3u_{xx}u_x+u_{xxx}u=0.$ One is stable, the other not. Finally, we extend the equation $u_t=\pa^5_x u^{-{2/3}}$ to a large class of equations $ u_t=\partial_x^l u^{-m/n}, l\ge1, n\not=0, m,n \in \Z, $ which still have the singular cusp-like traveling wave solutions.