Mathematical Physics, Analysis and Geometry

, Volume 7, Issue 4, pp 289–308

# A New Integrable Hierarchy, Parametric Solutions and Traveling Wave Solutions

• Zhijun Qiao
• Shengtai Li
Article

## Abstract

In this paper we give a new integrable hierarchy. In the hierarchy there are the following representatives:
$$\begin{gathered} u_t = \partial _x^5 u^{{{ - 2} \mathord{\left/ {\vphantom {{ - 2} 3}} \right. \kern-\nulldelimiterspace} 3}} , \hfill \\ u_t = \partial _x^5 \frac{{\left( {u^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 3}} \right. \kern-\nulldelimiterspace} 3}} } \right)_{xx} - 2\left( {u^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 6}} \right. \kern-\nulldelimiterspace} 6}} } \right)_x^2 }}{u}, \hfill \\ u_{xxt} + 3u_{xx} u_x + u_{xxx} u_x = 0 \hfill \\ \end{gathered}$$
The first two are the positive members of the hierarchy, and the first equation was a reduction of an integrable (2+1)-dimensional system (see B. G. Konopelchenko and V. G. Dubrovsky, Phys. Lett. A102 (1984), 15–17). The third one is the first negative member. All nonlinear equations in the hierarchy are shown to have 3×3 Lax pairs through solving a key 3×3 matrix equation, and therefore they are integrable. Under a constraint between the potential function and eigenfunctions, the 3×3 Lax pair and its adjoint representation are nonlinearized to be two Liouville-integrable Hamiltonian systems. On the basis of the integrability of 6N-dimensional systems we give the parametric solution of all positive members in the hierarchy. In particular, we obtain the parametric solution of the equation u t =∂5 x u−2/3. Finally, we present the traveling wave solutions (TWSs) of the above three representative equations. The TWSs of the first two equations have singularities, but the TWS of the 3rd one is continuous. The parametric solution of the 5th-order equation u t =∂5 x u−2/3 can not contain its singular TWS. We also analyse Gaussian initial solutions for the equations u t =∂5 x u−2/3, and u xxt +3u xx u x +u xxx u=0. Both of them are stable.
Hamiltonian system matrix equation zero curvature representation parametric solution traveling wave solution

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## Authors and Affiliations

• Zhijun Qiao
• 1
• 2
• Shengtai Li
• 2
1. 1.Department of MathematicsUniversity of Texas–Pan AmericanEdinburgUSA
2. 2.Los Alamos National LaboratoryLos AlamosUSA