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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: \[ m_t=12(1/m^2)_{xxx}-12(1/m^2)_x \tag{1} \]
\[ m_t+m_x(u^2-u_x^2)+2m^2u_x=0,\quad m=u-u_{xx} \tag{2.} \] 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 \(ce^{-|x-ct|}\) (\(c\) is a constant).

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
92C20 Neural biology
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[1] DOI: 10.1002/sapm1974534249 · Zbl 0408.35068 · doi:10.1002/sapm1974534249
[2] DOI: 10.1137/1.9781611970883 · doi:10.1137/1.9781611970883
[3] DOI: 10.1007/978-1-4757-1693-1 · doi:10.1007/978-1-4757-1693-1
[4] Cao C. W., Sci. China, Ser. A: Math., Phys., Astron. 32 pp 701– (1989)
[5] Cao C. W., Sci. China, Ser. A: Math., Phys., Astron. 33 pp 528– (1990)
[6] DOI: 10.1103/PhysRevLett.71.1661 · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[7] Chen J. H., Brain Res. 846 pp 243– (1999) · doi:10.1016/S0006-8993(99)01883-1
[8] DOI: 10.1016/j.neuroscience.2004.04.021 · doi:10.1016/j.neuroscience.2004.04.021
[9] DOI: 10.1006/jfan.1997.3231 · Zbl 0907.35009 · doi:10.1006/jfan.1997.3231
[10] DOI: 10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D · Zbl 0940.35177 · doi:10.1002/(SICI)1097-0312(199908)52:8<949::AID-CPA3>3.0.CO;2-D
[11] DOI: 10.1002/cpa.3046 · Zbl 1038.76011 · doi:10.1002/cpa.3046
[12] DOI: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L · Zbl 1049.35149 · doi:10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L
[13] DOI: 10.1142/1109 · doi:10.1142/1109
[14] DOI: 10.1063/1.525495 · Zbl 0495.58016 · doi:10.1063/1.525495
[15] Fokas A. S., Lett. Nuovo Cimento Soc. Ital. Fis. 28 pp 299– (1980) · doi:10.1007/BF02798794
[16] DOI: 10.1007/978-3-642-77769-1 · doi:10.1007/978-3-642-77769-1
[17] DOI: 10.1016/0167-2789(81)90004-X · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X
[18] DOI: 10.1103/PhysRevLett.19.1095 · doi:10.1103/PhysRevLett.19.1095
[19] DOI: 10.1017/CBO9780511624056 · Zbl 0892.76001 · doi:10.1017/CBO9780511624056
[20] DOI: 10.1017/S0022112001007224 · Zbl 1037.76006 · doi:10.1017/S0022112001007224
[21] Lax P. D., Commun. Pure Appl. Math. 28 pp 141– (1975) · Zbl 0295.35004 · doi:10.1002/cpa.3160280105
[22] DOI: 10.1016/j.jde.2004.09.007 · Zbl 1082.35127 · doi:10.1016/j.jde.2004.09.007
[23] DOI: 10.1063/1.2365758 · Zbl 1112.37063 · doi:10.1063/1.2365758
[24] Qiao Z. J., Finite-dimensional Integrable System and Nonlinear Evolution Equations (2002)
[25] Qiao, Z. J. , Master thesis, Zhengzhou University, 1989.
[26] DOI: 10.1007/s00220-003-0880-y · Zbl 1020.37046 · doi:10.1007/s00220-003-0880-y
[27] DOI: 10.1209/epl/i2005-10453-y · doi:10.1209/epl/i2005-10453-y
[28] Tu G. Z., Northeast. Math. J. 6 pp 26– (1990)
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