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New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons. (English) Zbl 1152.81587

Summary: In this paper, we propose a new completely integrable hierarchy. Particularly in the hierarchy we draw two new soliton equations:

${m}_{t}=12{\left(1/{m}^{2}\right)}_{xxx}-12{\left(1/{m}^{2}\right)}_{x}\phantom{\rule{2.em}{0ex}}\left(1\right)$
${m}_{t}+{m}_{x}\left({u}^{2}-{u}_{x}^{2}\right)+2{m}^{2}{u}_{x}=0,\phantom{\rule{1.em}{0ex}}m=u-{u}_{xx}\phantom{\rule{2.em}{0ex}}\left(2·\right)$

The first one is the second positive member in the hierarchy while the second one is the second negative member in the hierarchy. Both equations can be derived from the two-dimensional Euler equation by using the approximation procedure. All equations in the hierarchy are proven to have bi-Hamiltonian operators and Lax pairs through solving a crucial matrix equation. Moreover, we develop parametric solutions of the entire hierarchy through constructing two kinds of constraints; one is for all negative members of the hierarchy on a symplectic submanifold, and the other is for all positive members in the standard symplectic space. The most interesting things are both equations possess new type of peaked solitons – continuous and piecewise smooth “W-/M-shape peak” soliton solutions. In addition, we find new cusp solitons-cuspons for the second equation and one-single-peak solitons for the first-which are also continuous and piecewise smooth but not in the regular type $c{e}^{-|x-ct|}$ ($c$ is a constant).

##### MSC:
 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 35Q51 Soliton-like equations 37K40 Soliton theory, asymptotic behavior of solutions 92C20 Neural biology