Geodesic
Unfamiliar aspects of Bäcklund transformations and an associated Degasperis–Procesi equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 3, pp. 449-464 Cet article a éte moissonné depuis la source Math-Net.Ru

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We summarize the results of our recent work on Bäcklund transformations (BTs), particularly focusing on the relation between BTs and infinitesimal symmetries. We present a BT for an associated Degasperis–Procesi (aDP) equation and its superposition principle and investigate the solutions generated by applying this BT. Following our general methodology, we use the superposition principle of the BT to generate the infinitesimal symmetries of the aDP equation.
Keywords: Bäcklund transformation, Degasperis–Procesi equation, symmetry.
Mots-clés : superposition principle, soliton 
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A. G. Rasin; J. Schiff. Unfamiliar aspects of Bäcklund transformations and an associated Degasperis–Procesi equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 3, pp. 449-464. http://geodesic.mathdoc.fr/item/TMF_2018_196_3_a6/

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

[2] P. Olver, Prilozheniya grupp Li k differentsialnym uravneniyam, Mir, M., 1989 | MR

[3] P. J. Olver, “Evolution equations possessing infinitely many symmetries”, J. Math. Phys., 18:6 (1977), 1212–1215 | DOI | MR

[4] S. Kumei, “Invariance transformations, invariance group transformations, and invariance groups of the sine-Gordon equations”, J. Math. Phys., 16:12 (1975), 2461–2468 | DOI | MR

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

[6] J. Schiff, “The Camassa–Holm equation: a loop group approach”, Phys. D, 121:1–2 (1998), 24–43 | DOI | MR

[7] A. G. Rasin, J. Schiff, “Bäcklund transformations for the Boussinesq equation and merging solitons”, J. Phys. A: Math. Theor., 50:32 (2017), 325202, 21 pp. | DOI | MR

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

[9] V. E. Adler, A. B. Shabat, “Toward a theory of integrable hyperbolic equations of third order”, J. Phys. A: Math. Theor., 45:39 (2012), 395207, 17 pp. | DOI | MR

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

[11] H. R. Dullin, G. A. Gottwald, D. D. Holm, “Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves”, Fluid Dynam. Res., 33:1–2 (2003), 73–95 | DOI | MR

[12] D. D. Holm, M. F. Staley, “Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a $1+1$ nonlinear evolutionary PDE”, Phys. Lett. A, 308:5–6 (2003), 437–444 | DOI | MR

[13] D. D. Holm, M. F. Staley, “Wave structure and nonlinear balances in a family of evolutionary PDEs”, SIAM J. Appl. Dyn. Syst., 2:3 (2003), 323–380 | DOI | MR

[14] A. N. W. Hone, J. P. Wang, “Prolongation algebras and Hamiltonian operators for peakon equations”, Inverse Problems, 19:1 (2003), 129–145 | DOI | MR

[15] M. C. Nucci, “Painlevé property and pseudopotentials for nonlinear evolution equations”, J. Phys. A: Math. Gen., 22:15 (1989), 2897–2914 | DOI | MR

[16] J. Kang, X. Liu, P. J. Olver, C. Qu, “Liouville correspondences between integrable hierarchies”, SIGMA, 13 (2017), 035, 26 pp. | DOI | MR

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

[18] V. O. Vakhnenko, E. J. Parkes, “Periodic and solitary-wave solutions of the Degasperis–Procesi equation”, Chaos Solitons Fractals, 20:5 (2004), 1059–1073 | DOI | MR

[19] J. Lenells, “Traveling wave solutions of the Degasperis–Procesi equation”, J. Math. Anal. Appl., 306:1 (2005), 72–82 | DOI | MR

[20] Y. Matsuno, “Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit”, Inverse Problems, 21:5 (2005), 1553–1570 | DOI | MR

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

[22] B.-F. Feng, K.-I. Maruno, Y. Ohta, “On the $\tau$-functions of the Degasperis–Procesi equation”, J. Phys. A: Math. Theor., 46:4 (2013), 045205, 25 pp. | DOI | MR

[23] B.-F. Feng, K.-I. Maruno, Y. Ohta, “An integrable semi-discrete Degasperis–Procesi equation”, Nonlinearity, 30:6 (2017), 2246–2267 | DOI | MR

[24] S. Stalin, M. Senthilvelan, “Multi-loop soliton solutions and their interaction in the Degasperis–Procesi equation”, Phys. Scr., 86:1 (2012), 015006, arXiv: 1207.4634 | DOI

[25] A. B. Shabat, V. E. Adler, V. G. Marikhin, A. A. Mikhailov, V. V. Sokolov (red.), Entsiklopediya integriruemykh sistem, ITF, M., 2009

[26] A. P. Fordy, J. Gibbons, “Some remarkable nonlinear transformations”, Phys. Lett. A, 75:5 (1980), 325 | DOI | MR

[27] A. P. Fordy, J. Gibbons, “Factorization of operators. I. Miura transformations”, J. Math. Phys., 21:10 (1980), 2508–2510 | DOI | MR

[28] A. P. Fordy, J. Gibbons, “Factorization of operators. II”, J. Math. Phys., 22:6 (1981), 1170–1175 | DOI | MR

[29] V. Novikov, “Generalizations of the Camassa–Holm equation”, J. Phys. A: Math. Theor., 42:34 (2009), 342002, 14 pp. | DOI | MR

[30] A. N. W. Hone, J. P. Wang, “Integrable peakon equations with cubic nonlinearity”, J. Phys. A: Math. Theor., 41:37 (2008), 372002, 10 pp. | DOI | MR

[31] W. Fu, F. Nijhoff, “On reductions of the discrete Kadomtsev–Petviashvili-type equations”, J. Phys. A: Math. Theor., 50:50 (2017), 505203, arXiv: 1705.04819 | DOI