2002/09/11 by Darryl D. Holm, Holm, Darryl D., Zhijun Qiao +1
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #nlin.SI
paper · pdf · doi:10.48550/arxiv.nlin/0209026
22 pages, 8 figures
arxiv created 2002/11/01 · arxiv updated 2009/11/30
This paper gives an integrable hierarchy of nonlinear evolution equations. In this hierarchy there are the following representative equations: \beqq & & ut=\pa5x u^-2/3, & & ut=\pa5x\frac(u^-1/3)xx -2(u^-1/6)x2u; & & uxxt+3uxxux+uxxxu=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× 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=\pa5x 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=\pa5x u^-2/3, and uxxt+3uxxux+uxxxu=0. One is stable, the other not. Finally, we extend the equation ut=\pa5x u^-2/3 to a large class of equations ut=∂xl u-m/n, l≥1, n\not=0, m,n ∈ \Z, which still have the singular cusp-like traveling wave solutions.