Geodesic
Bäcklund transformations for the Degasperis–Procesi equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 203 (2020) no. 3, pp. 365-379 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the Bäcklund transformation for the Degasperis–Procesi (DP) equation. Using the reciprocal transformation and the associated DP equation, we construct the Bäcklund transformation for the DP equation involving both dependent and independent variables. We also obtain the corresponding nonlinear superposition, which we use together with the Bäcklund transformation to derive some soliton solutions of the DP equation.
Keywords: Degasperis–Procesi equation, Bäcklund transformation, nonlinear superposition formula
Mots-clés : soliton. 
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Hui Mao; Gaihua Wang. Bäcklund transformations for the Degasperis–Procesi equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 203 (2020) no. 3, pp. 365-379. http://geodesic.mathdoc.fr/item/TMF_2020_203_3_a3/

[1] A. Degasperis, M. Procesi, “Asymptotic integrability”, Symmetry and Perturbation Theory (Rome, Italy, December 16–22, 1998), eds. A. Degasperis, G. Gaeta, World Sci., Singapore, 1999, 23–37 | MR | Zbl

[2] A. Degasperis, D. D. Kholm, A. Khon, “Novoe integriruemoe uravnenie s pikonnymi resheniyami”, TMF, 133:2 (2002), 170–183 | DOI | DOI | MR

[3] R. Camassa, D. D. Holm, “An integrable shallow water equation with peaked solitons”, Phys. Rev. Lett., 71:11 (1993), 1661–1664, arXiv: patt-sol/9305002 | DOI | MR

[4] A. Constantin, R. I. Ivanov, J. Lenells, “Inverse scattering transform for the Degasperis–Procesi equation”, Nonlinearity, 23:10 (2010), 2559–2575, arXiv: 1205.4754 | DOI | MR

[5] H. Lundmark, J. Szmigielski, “Multi-peakon solutions of the Degasperis–Procesi equation”, Inverse Problems, 19:6 (2003), 1241–1245, arXiv: nlin/0503033 | DOI | MR

[6] H. Lundmark, J. Szmigielski, “Degasperis–Procesi peakons and the discrete cubic string”, Internat. Math. Res. Rap., 2005:2 (2005), 53–116 | DOI | MR

[7] Y. Matsuno, “The $N$-soliton solution of the Degasperis–Procesi equation”, Inverse Probl., 21:6 (2005), 2085–2101 | DOI | MR | Zbl

[8] Y. Matsuno, “Multisolitons of the Degasperis–Procesi equation and their peakon limit”, Inverse Probl., 21:5 (2005), 1553–1570, arXiv: nlin/0511029 | DOI | MR

[9] Y. Matsuno, “Cusp and loop soliton solutions of short-wave models for the Camassa–Holm and Degasperis–Procesi equations”, Phys. Lett. A, 359:5 (2006), 451–457 | DOI | MR

[10] A. Constantin, R. Ivanov, “Dressing method for the Degasperis–Procesi equation”, Stud. Appl. Math., 138:2 (2017), 205–226 | DOI | MR

[11] C. Rogers, W. F. Shadwick, Bäcklund Transformations and Their Applications, Mathematics in Science and Engineering, 161, Academic Press, New York, 1982 | MR

[12] C. Rogers, W. K. Schief, Bäcklund and Darboux transformations. Geometry and Modern Applications in Soliton Theory, Cambridge Texts in Applied Mathematics, 30, Cambridge Univ. Press, Cambridge, 2002 | DOI | MR

[13] C. Gu, H. Hu, Z. Zhou, Darboux Transformation in Soliton Theory and Its Geometric Applications, Shanghai Sci. Tech. Press, Shanghai, 2005

[14] J. Hietarinta, N. Joshi, F. W. Nijhoff, Discrete Systems and Integrability, Cambridge Texts in Applied Mathematics, 54, Cambridge Univ. Press, Cambridge, 2016 | DOI | MR

[15] Y. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach, Progress in Mathematics, 219, Birkhäuser, Basel, 2003 | DOI | MR

[16] D. Levi, R. Benguria, “Bäcklund transformations and nonlinear differential difference equations”, Proc. Nat. Acad. Sci. USA, 77:9 (1980), 5025–5027 | DOI | MR

[17] D. Levi, “Nonlinear differential difference equations as Bäcklund transformations”, J. Phys. A: Math. Gen., 14:5 (1981), 1083–1098 | DOI | MR

[18] A. G. Rasin, J. Schiff, “The Gardner method for symmetries”, J. Phys. A: Math. Theor., 46:15 (2013), 155202, 15 pp. | DOI | MR

[19] A. G. Rasin, J. Schiff, “Bäcklund transformations for the Camassa–Holm equation”, J. Nonlinear Sci., 27:1 (2017), 45–69 | DOI | MR

[20] G. Wang, Q. P. Liu, H. Mao, “The modified Camassa-Holm equation: Bäcklund transformations and nonlinear superposition formulae”, accepted, J. Phys. A: Math. Theor.

[21] A. G. Rasin, D. Shiff, “Neizvestnye aspekty preobrazovanii Beklunda i assotsiirovannoe uravnenie Degasperisa–Prochezi”, TMF, 196:3 (2018), 449–464 | DOI | DOI

[22] M. C. Nucci, “Painlevé property and pseudopotentials for nonlinear evolution equations”, J. Phys. A: Math. Gen., 22:15 (1989), 2897–2913 | DOI | MR

[23] M. Musette, R. Conte, “Bäcklund transformation of partial differential equations from the Painlevé–Gambier classication. I. Kaup–Kupershmidt equation”, J. Math. Phys., 39:10 (1998), 5617–5630 | DOI | MR

[24] M. Musette, C. Verhoeven, “Nonlinear superposition formula for the Kaup–Kupershmidt partial differential equation”, Phys. D, 144:1–2 (2000), 211–220 | DOI | MR

[25] C. Rogers, P. Wong, “On reciprocal Bäcklund transformations of inverse scattering schemes”, Phys. Scr., 30:1 (1984), 10–14 | DOI | MR