Multisoliton solutions of the two-component Camassa–Holm equation and its reductions

Gaihua Wang

Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 3, pp. 359-386

Résumé

The Bäcklund transformation for an integrable two-component Camassa–Holm ($2$CH) equation is presented and studied. It involves both dependent and independent variables. A nonlinear superposition formula is given for constructing multisoliton, multiloop, and multikink solutions of the $2$CH equation. We also present solutions of the Camassa–Holm equation, the two-component Hunter–Saxton ($2$HS) equation, and the Hunter–Saxton equation, which all arise from solutions of the $2$CH equation. By appropriate limit procedures, a solution of the $2$HS equation is successfully obtained from that of the $2$CH equation, which is worked out with the method of Bäcklund transformations. By analyzing the solution, we obtain the soliton and loop solutions for $2$HS equation.

Détail

Keywords: two-component Camassa–Holm equation, two-component Hunter–Saxton equation, Bäcklund transformation, reduction.
Mots-clés : soliton

Bibliographie

[1] S.-Q. Liu, Y. J. Zhang, “Deformation of semisimple bihamiltonian structures of hydrodynamic type”, J. Geom. Phys., 54:4 (2005), 427–453 | DOI | MR

[2] G. Falqui, On a two-component generalization of the CH equation, Talk given at the conference “Analytic and Geometric Theory of the Camassa–Holm Equation and Integrable Systems” (Bologna, September 22–25, 2004)

[3] J. Schiff, “Zero curvature formulations of dual hierarchies”, J. Math. Phys., 37:4 (1996), 1928–1938 | DOI | MR

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

[5] A. S. Fokas, “On a class of physically important integrable equations”, Phys. D, 87:1–4 (1995), 145–150 | DOI | MR

[6] B. Fuchssteiner, A. S. Fokas, “Symplectic structures, their Bäcklund transformations and hereditary symmetries”, Phys. D, 4:1 (1981), 47–66 | DOI | MR

[7] A. Parker, “On the Camassa–Holm equation and a direct method of the solution. I. Bilinear form and solitary waves”, Proc. Roy. Soc. London Ser. A, 460:2050 (2004), 2929–2957 | DOI | MR | Zbl

[8] A. Parker, “On the Camassa–Holm equation and a direct method of the solution. II. Soliton solutions”, Proc. Roy. Soc. London Ser. A, 461:2063 (2005), 3611–3632 | DOI | MR

[9] Y. S. Li, J. E. Zhang, “The multiple-soliton solution of the Camassa–Holm equation”, Proc. Roy. Soc. London Ser. A, 460:2049 (2004), 2617–2627 | DOI | MR | Zbl

[10] Y. Matsuno, “Parametric representation for the multisoliton solution of the Camassa–Holm equation”, J. Phys. Soc. Japan, 74:7 (2005), 1983–1987 | DOI | MR

[11] Y. Matsuno, “Multisoliton solutions of the two-component Camassa–Holm system and their reductions”, J. Phys. A: Math. Theor., 50:34 (2017), 345202, 28 pp. | DOI | MR

[12] B. Q. Xia, R. G. Zhou, Z. J. Qiao, “Darboux transformation and multi-soliton solutions of the Camassa–Holm equation and modified Camassa–Holm equation”, J. Math. Phys., 57:10 (2016), 103502, 12 pp. | DOI | MR

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

[14] A. Constantin, “On the scattering problem for the Camassa–Holm equation”, Proc. Roy. Soc. London Ser. A, 457:2008 (2001), 953–970 | DOI | MR

[15] R. Beals, D. H. Sattinger, J. Szmigielski, “Acoustic scattering and the extended Korteweg–de Vries hierarchy”, Adv. Math., 140:2 (1998), 190–206 | DOI | MR

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

[17] A. Constantin, W. A. Strauss, “Stability of peakons”, Commun. Pure Appl. Math., 53:5 (2000), 603–610 | 3.0.CO;2-L class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[18] A. Constantin, W. A. Strauss, “Stability of the Camassa–Holm solitons”, J. Nonlinear Sci., 12:4 (2002), 415–422 | DOI | MR

[19] R. S. Johnson, “On solutions of the Camassa–Holm equation”, Proc. Roy. Soc. London Ser. A, 459:2035 (2003), 1687–1708 | DOI | MR

[20] P. J. Olver, P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support”, Phys. Rev. E, 53:2 (1996), 1900–1906 | DOI | MR

[21] D. D. Holm, R. I. Ivanov, “Two-component CH system: inverse scattering, peakons and geometry”, Inverse Problems, 27:4 (2011), 045013, 19 pp. | DOI | MR

[22] A. Constantin, R. I. Ivanov, “On an integrable two-component Camassa–Holm shallow water system”, Phys. Lett. A, 372:48 (2008), 7129–7132 | DOI | MR

[23] M. J. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Applied Mathematics, 4, SIAM, Philadelphia, PA, 1981 | MR

[24] J. Escher, O. Lechtenfeld, Z. Y. Yin, “Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation”, Discrete Continuous Dyn. Syst., 19:3 (2007), 493–513 | DOI | MR

[25] G. L. Gui, Y. Liu, “On the global existence and wave-breaking criteria for the two-component Camassa–Holm system”, J. Funct. Anal., 258:12 (2010), 4251–4278 | DOI | MR

[26] J. B. Li, Y. S. Li, “Bifurcations of travelling wave solutions for a two-component Camassa–Holm equation”, Acta. Math. Sin. (English Ser.), 24:8 (2008), 1319–1330 | DOI | MR

[27] M. Chen, S.-Q. Liu, Y. J. Zhang, “A two-component generalization of the Camassa–Holm equation and its solutions”, Lett. Math. Phys., 75:1 (2006), 1–15 | DOI | MR

[28] C.-Z. Wu, “On solutions of the two-component Camassa–Holm system”, J. Math. Phys., 47:8 (2006), 083513, 11 pp. | DOI | MR

[29] J. Lin, B. Ren, H.-M. Li, Y.-S. Li, “Soliton solutions for two nonlinear partical differential equations using a Darboux transformation of the Lax pairs”, Phys. Rev. E, 77:3 (2008), 036605, 10 pp. | DOI | MR

[30] Yui-Tsin Yao, E-Khuei Khuan, Yun-Bo Tszen, “Dvukhkomponentnoe uravnenie Kamassy–Kholma s samosoglasovannymi istochnikami i ego mnogosolitonnye resheniya”, TMF, 162:1 (2010), 75–86 | DOI | DOI | MR | Zbl

[31] B. Q. Xia, Z. J. Qiao, “A new two-component integrable system with peakon solutions”, Proc. Roy. Soc. London Ser. A, 471:2175 (2015), 20140750, 20 pp. | DOI | MR

[32] Q. Y. Hu, Z. Y. Yin, “Well-posedness and blow-up phenomena for a periodic two-component Camassa–Holm equation”, Proc. Roy. Soc. Edinburgh Sect. A, 141:1 (2011), 93–107 | DOI | MR

[33] M. Chen, S.-Q. Liu, Y. J. Zhang, “Hamiltonian structures and their reciprocal transformations for the $r$-KdV-CH hierarchy”, J. Geom. Phys., 59:9 (2009), 1227–1243 | DOI | MR

[34] G. Falqui, “On a Camassa–Holm type equation with two dependent variables”, J. Phys. A: Math. Gen., 39:2 (2006), 327–342 | DOI | MR

[35] H. Aratyn, J. F. Gomes, A. H. Zimerman, “On a negative flow of the AKNS hierarchy and its relation to a two-component Camassa–Holm equation”, SIGMA, 2 (2006), 070, 12 pp. | DOI | MR | Zbl

[36] H. Aratyn, J. F. Gomes, A. H. Zimerman, “On negative flows of the AKNS hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type”, J. Phys. A: Math. Gen., 39:5 (2006), 1099–1114 | DOI | MR

[37] J. B. Li, Z. J. Qiao, “Peakon, pseudo-peakon, and cuspon solutions for two generalized Camassa–Holm equations”, J. Math. Phys., 54:12 (2013), 123501, 14 pp. | DOI | MR

[38] Z. G. Guo, Y. Zhou, “On solutions to a two-component generalized Camassa–Holm equation”, Stud. Appl. Math., 124:3 (2010), 307–322 | DOI | MR

[39] M. V. Pavlov, “The Gurevich–Zybin system”, J. Phys. A: Math. Gen., 38:17 (2005), 3823–3840 | DOI | MR

[40] S. Y. Lou, B.-F. Feng, R. X. Yao, “Multi-soliton solution to the two-component Hunter–Saxton equation”, Wave Motion, 65 (2016), 17–28 | DOI | MR

[41] L. Yan, J.-F. Song, C.-Z. Qu, “Nonlocal symmetries and geometric integrability of multi-component Camassa–Holm and Hunter–Saxton systems”, Chinese Phys. Lett., 28:5 (2011), 050204, 5 pp. | DOI

[42] C. X. Guan, Z. Y. Yin, “Global weak solutions and smooth solutions for a two-component Hunter–Saxton system”, J. Math. Phys., 52:10 (2011), 103707, 9 pp. | DOI | MR

[43] J. J. Liu, Z. Y. Yin, “Global weak solutions for a periodic two-component $\mu$-Hunter–Saxton system”, Monatsh. Math., 168:3–4 (2012), 503–521 | DOI | MR

[44] B. Moon, Y. Liu, “Wave breaking and global existence for the generalized periodic two-component Hunter–Saxton system”, J. Differ. Eq., 253:1 (2012), 319–355 | DOI | MR

[45] D. F. Zuo, “A two-component $\mu$-Hunter–Saxton equation”, Inverse Problems, 26:8 (2010), 085003, 9 pp. | DOI | MR

[46] C. H. Li, S. Q. Wen, A. Y. Chen, “Single peak solitary wave and compacton solutions of the generalized two-component Hunter–Saxton system”, Nonlinear Dyn., 79:2 (2015), 1575–1585 | DOI | MR

[47] B. Moon, “Solitary wave solutions of the generalized two-component Hunter–Saxton system”, Nonlinear Anal., 89 (2013), 242–249 | DOI | MR

[48] J. K. Hunter, R. Saxton, “Dynamics of director fields”, SIAM J. Appl. Math., 51:6 (1991), 1498–1521 | DOI | MR

[49] G. H. Wang, Q. P. Liu, H. Mao, “The modified Camassa–Holm equation: Bäcklund transformation and nonlinear superposition formula”, J. Phys. A: Math. Theor., 53:29 (2020), 294003, 15 pp. | DOI | MR

[50] Khuei Mao, Gai-Khua Van, “Preobrazovanie Beklunda dlya uravneniya Degasperisa–Prochezi”, TMF, 203:3 (2020), 365–379 | DOI | DOI

[51] H. Mao, Q. P. Liu, “The short pulse equation: Bäcklund transformations and applications”, Stud. Appl. Math., 145:4 (2020), 791–811 | DOI | MR

[52] M. Xue, Q. P. Liu, H. Mao, “Bäcklund transformations for the modified short pulse equation and complex modified short pulse equation”, Eur. Phys. J. Plus, 137 (2022), 500 | DOI

[53] R. Hirota, “Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons”, Phys. Rev. Lett., 27:18 (1971), 1192–1194 | DOI

[54] R. Hirota, The Direct Method in Soliton Theory, Translated from the 1992 Japanese original, Cambridge Tracts in Mathematics, 155, eds. A. Nagai, J. Nimmo, C. Gilson, Cambridge Univ. Press, Cambridge, 2004 | DOI | MR