Frontiers of Mathematics in China >
Integrable peakon systems with weak kink and kink-peakon interactional solutions
Received date: 18 Apr 2013
Accepted date: 03 Jun 2013
Published date: 01 Oct 2013
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We report two integrable peakon systems that have weak kink and kink-peakon interactional solutions. Both peakon systems are guaranteed integrable through providing their Lax pairs. The peakon and multi-peakon solutions of both equations are studied. In particular, the two-peakon dynamic systems are explicitly presented and their collisions are investigated. The weak kink solution is studied, and more interesting, the kink-peakon interactional solutions are proposed for the first time.
Key words: Integrable system; Lax pair; peakon; weak kink; kink-peakon
Zhijun QIAO , Baoqiang XIA . Integrable peakon systems with weak kink and kink-peakon interactional solutions[J]. Frontiers of Mathematics in China, 2013 , 8(5) : 1185 -1196 . DOI: 10.1007/s11464-013-0314-x
1 |
Beals R, Sattinger D, Szmigielski J. Acoustic scattering and the extended Korteweg de Vries hierarchy. Adv Math, 1998, 140: 190-206
|
2 |
Camassa R, Holm D D. An integrable shallow water equation with peaked solitons. Phys Rev Lett, 1993, 71: 1661-1664
|
3 |
Camassa R, Holm D D, Hyman J M. A new integrable shallow water equation. Adv Appl Mech, 1994, 31: 1-33
|
4 |
Constantin A. On the inverse spectral problem for the Camassa-Holm equation. J Funct Anal, 1998, 155: 352-363
|
5 |
Constantin A, Gerdjikov V S, Ivanov R I. Inverse scattering transform for the Camassa- Holm equation. Inverse Problems, 2006, 22: 2197-2207
|
6 |
Constantin A, Strauss W A. Stability of peakons. Comm Pure Appl Math, 2000, 53: 603-610
|
7 |
Degasperis A, Holm D D, Hone A N W. A new integrable equation with peakon solutions. Theoret Math Phys, 2002, 133: 1463-1474
|
8 |
Degasperis A, Procesi M. Asymptotic Integrability. In: Degasperis A, Gaeta G, eds. Symmetry and Perturbation Theory. Singapore: World Scientific, 1999, 23-37
|
9 |
Dullin H R, Gottwald G A, Holm D D. An integrable shallow water equation with linear and nonlinear dispersion. Phys Rev Lett, 2001, 87: 194501
|
10 |
Fokas A S. On a class of physically important integrable equations. Physica D, 1995, 87: 145-150
|
11 |
Fokas A S, Liu Q M. Asymptotic integrability of water waves. Phys Rev Lett, 1996, 77: 2347-2351
|
12 |
Fuchssteiner B. Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation. Physica D, 1996, 95: 229-243
|
13 |
Fuchssteiner B, Fokas A S. Symplectic structures, their Baecklund transformations and hereditary symmetries. Physica D, 1981, 4: 47-66
|
14 |
Gesztesy F, Holden H. Algebro-geometric solutions of the Camassa-Holm hierarch. Rev Mat Iberoam, 2003, 19: 73-142
|
15 |
Gui G L, Liu Y, Olver P J, Qu C Z. Wave-breaking and peakons for a modified Camassa- Holm equation. Comm Math Phys, 2013, 319: 731-759
|
16 |
Hone A N W, Wang J P. Integrable peakon equations with cubic nonlinearity. J Phys A: Math Theor, 2008, 41: 372002
|
17 |
Lorenzoni P, Pedroni M. On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations. Int Math Res Not, 2004, 75: 4019-4029
|
18 |
Lundmark H. Formation and dynamics of shock waves in the Degasperis-Procesi equation. J Nonlinear Sci, 2007, 17: 169-198
|
19 |
Lundmark H, Szmigielski J. Multi-peakon solutions of the Degasperis-Procesi equation. Inverse Problems, 2003, 19: 1241-1245
|
20 |
Novikov V. Generalizations of the Camassa-Holm equation. J Phys A: Math Theor, 2009, 42: 342002
|
21 |
Olver P J, Rosenau P. Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys Rev E, 1996, 53: 1900-1906
|
22 |
Qiao Z J. The Camassa-Holm hierarchy, N-dimensional integrable systems, and algebro- geometric solution on a symplectic submanifold. Comm Math Phys, 2003, 239: 309-341
|
23 |
Qiao Z J. A new integrable equation with cuspons and W/M-shape-peaks solitons. J Math Phys, 2006, 47: 112701
|
24 |
Qiao Z J. New integrable hierarchy, parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solutions. J Math Phys, 2007, 48: 082701
|
25 |
Qiao Z J, Li X Q. An integrable equation with nonsmooth solitons. Theoret Math Phys, 2011, 167: 584-589
|
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