Integrable Semi-Discretization for a Modified Camassa–Holm Equation with Cubic Nonlinearity

Bao-Feng Feng ; Heng-Chun Hu ; Han-Han Sheng ; Wei Yin ; Guo-Fu Yu

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

In the present paper, an integrable semi-discretization of the modified Camassa–Holm (mCH) equation with cubic nonlinearity is presented. The key points of the construction are based on the discrete Kadomtsev–Petviashvili (KP) equation and appropriate definition of discrete reciprocal transformations. First, we demonstrate that these bilinear equations and their determinant solutions can be derived from the discrete KP equation through Miwa transformation and some reductions. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solutions in the Gram-type determinant form. Finally, we obtain an integrable semi-discrete analog of the mCH equation by introducing dependent variables and discrete reciprocal transformation. It is also shown that the semi-discrete mCH equation converges to the continuous one in the continuum limit.

Détail

Keywords: modified Camassa–Holm equation, discrete KP equation
Mots-clés : Miwa transformation.

Bibliographie

[1] Bies P.M., Górka P., Reyes E.G., “The dual modified Korteweg–de Vries–Fokas–Qiao equation: geometry and local analysis”, J. Math. Phys., 53 (2012), 073710, 19 pp. | DOI

[2] Boutet de Monvel A., Karpenko I., Shepelsky D., “A Riemann–Hilbert approach to the modified Camassa–Holm equation with nonzero boundary conditions”, J. Math. Phys., 61 (2020), 031504, 24 pp., arXiv: 1911.07263 | DOI

[3] Chang X., Szmigielski J., “Lax integrability of the modified Camassa–Holm equation and the concept of peakons”, J. Nonlinear Math. Phys., 23 (2016), 563–572, arXiv: 1610.06537 | DOI

[4] Chang X., Szmigielski J., “Liouville integrability of conservative peakons for a modified CH equation”, J. Nonlinear Math. Phys., 24 (2017), 584–595, arXiv: 1707.04989 | DOI

[5] Chang X., Szmigielski J., “Lax integrability and the peakon problem for the modified Camassa–Holm equation”, Comm. Math. Phys., 358 (2018), 295–341, arXiv: 1705.06451 | DOI

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

[7] Chen R.M., Liu Y., Qu C., Zhang S., “Oscillation-induced blow-up to the modified Camassa–{H}olm equation with linear dispersion”, Adv. Math., 272 (2015), 225–251 | DOI

[8] Feng B.-F., “Complex short pulse and coupled complex short pulse equations”, Phys. D, 297 (2015), 62–75, arXiv: 1312.6431 | DOI

[9] Feng B.-F., Maruno K., Ohta Y., “Integrable discretizations for the short wave model of the Camassa–Holm equation”, J. Phys. A, 43 (2010), 265202, 14 pp., arXiv: 1002.3649 | DOI

[10] Feng B.-F., Maruno K., Ohta Y., “Integrable discretizations of the short pulse equation”, J. Phys. A, 43 (2010), 085203, 14 pp., arXiv: 0912.1914 | DOI

[11] Feng B.-F., Maruno K., Ohta Y., “Integrable semi-discretization of a multi-component short pulse equation”, J. Math. Phys., 56 (2015), 043502, 15 pp., arXiv: 1504.00878 | DOI

[12] Feng B.-F., Maruno K., Ohta Y., “An integrable semi-discrete Degasperis–Procesi equation”, Nonlinearity, 30 (2017), 2246–2267, arXiv: 1510.03010 | DOI

[13] Feng B.-F., Sheng H.-H., Yu G.-F., “Integrable semi-discretizations and self-adaptive moving mesh method for a generalized sine-Gordon equation”, Numer. Algorithms, 94 (2023), 351–370 | DOI

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

[15] Fu W., Nijhoff F.W., “On reductions of the discrete Kadomtsev–Petviashvili-type equations”, J. Phys. A, 50 (2017), 505203, 21 pp., arXiv: 1705.04819 | DOI

[16] Fu Y., Gui G., Liu Y., Qu C., “On the Cauchy problem for the integrable modified Camassa–Holm equation with cubic nonlinearity”, J. Differential Equations, 255 (2013), 1905–1938, arXiv: 1108.5368 | DOI

[17] Fuchssteiner B., “Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa–Holm equation”, Phys. D, 95 (1996), 229–243 | DOI

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

[19] Gao Y., “On conservative sticky peakons to the modified Camassa–Holm equation”, J. Differential Equations, 365 (2023), 486–520, arXiv: 2211.03245 | DOI

[20] Gao Y., Li L., Liu J.-G., “A dispersive regularization for the modified Camassa–Holm equation”, SIAM J. Math. Anal., 50 (2018), 2807–2838, arXiv: 1707.06377 | DOI

[21] Gao Y., Li L., Liu J.-G., “Patched peakon weak solutions of the modified Camassa–Holm equation”, Phys. D, 390 (2019), 15–35, arXiv: 1703.07466 | DOI

[22] Gui G., Liu Y., Olver P.J., Qu C., “Wave-breaking and peakons for a modified Camassa–Holm equation”, Comm. Math. Phys., 319 (2013), 731–759 | DOI

[23] Hietarinta J., Joshi N., Nijhoff F.W., Discrete systems and integrability, Cambridge Texts Appl. Math., Cambridge University Press, Cambridge, 2016 | DOI

[24] Himonas A.A., Mantzavinos D., “The Cauchy problem for the Fokas–Olver–Rosenau–Qiao equation”, Nonlinear Anal., 95 (2014), 499–529 | DOI

[25] Hirota R., “Discrete analogue of a generalized Toda equation”, J. Phys. Soc. Japan, 50 (1981), 3785–3791 | DOI

[26] Hou Y., Fan E., Qiao Z., “The algebro-geometric solutions for the Fokas–Olver–Rosenau–Qiao (FORQ) hierarchy”, J. Geom. Phys., 117 (2017), 105–133 | DOI

[27] Hu H., Yin W., Wu H., “Bilinear equations and new multi-soliton solution for the modified Camassa–Holm equation”, Appl. Math. Lett., 59 (2016), 18–23 | DOI

[28] Kang J., Liu X., Olver P.J., Qu C., “Liouville correspondence between the modified KdV hierarchy and its dual integrable hierarchy”, J. Nonlinear Sci., 26 (2016), 141–170 | DOI

[29] Li J., Liu Y., “Stability of solitary waves for the modified Camassa–Holm equation”, Ann. PDE, 7 (2021), 14, 35 pp. | DOI

[30] Liu X., Liu Y., Qu C., “Orbital stability of the train of peakons for an integrable modified Camassa–Holm equation”, Adv. Math., 255 (2014), 1–37 | DOI

[31] Liu Y., Olver P.J., Qu C., Zhang S., “On the blow-up of solutions to the integrable modified Camassa–Holm equation”, Anal. Appl. (Singap.), 12 (2014), 355–368 | DOI

[32] Luo Z., Qiao Z., Yin Z., “On the Cauchy problem for a modified Camassa–Holm equation”, Monatsh. Math., 193 (2020), 857–877 | DOI

[33] Ma R., Zhang Y., Xiong N., Feng B.-F., “Short wave limit of the Novikov equation and its integrable semi-discretizations”, J. Phys. A, 54 (2021), 495701, 17 pp. | DOI

[34] Matsuno Y., “Bäcklund transformation and smooth multisoliton solutions for a modified Camassa–Holm equation with cubic nonlinearity”, J. Math. Phys., 54 (2013), 051504, 14 pp., arXiv: 1302.0107 | DOI

[35] Matsuno Y., “Smooth and singular multisoliton solutions of a modified Camassa–Holm equation with cubic nonlinearity and linear dispersion”, J. Phys. A, 47 (2014), 125203, 25 pp., arXiv: 1310.4011 | DOI

[36] Mikhailov A.V., “Integrability of the two-dimensional Thirring model”, JETP Lett., 23 (1976), 320–323

[37] Miwa T., “On Hirota's difference equations”, Proc. Japan Acad. Ser. A Math. Sci., 58 (1982), 9–12 | DOI

[38] Ohta Y., Hirota R., Tsujimoto S., Imai T., “Casorati and discrete Gram type determinant representations of solutions to the discrete KP hierarchy”, J. Phys. Soc. Japan, 62 (1993), 1872–1886 | DOI

[39] Ohta Y., Maruno K., Feng B.-F., “An integrable semi-discretization of the Camassa–Holm equation and its determinant solution”, J. Phys. A, 41 (2008), 355205, 30 pp., arXiv: 0805.2843 | DOI

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

[41] Qiao Z., “A new integrable equation with cuspons and W/M-shape-peaks solitons”, J. Math. Phys., 47 (2006), 112701, 9 pp. | DOI

[42] Qiao Z., Li X., An integrable equation with nonsmooth solitons,, Theoret. and Math. Phys., 167 (2011), 584–589 | DOI

[43] Qu C., Fu Y., Liu Y., “Well-posedness, wave breaking and peakons for a modified $\mu$-Camassa–Holm equation”, J. Funct. Anal., 266 (2014), 433–477 | DOI

[44] Sheng H.-H., Feng B.-F., Yu G.-F., “A generalized sine-Gordon equation: reductions and integrable discretizations”, J. Nonlinear Sci., 34 (2024), 55, 53 pp. | DOI

[45] Sheng H.-H., Yu G.-F., Feng B.-F., “An integrable semi-discretization of the modified Camassa–Holm equation with linear dispersion term”, Stud. Appl. Math., 149 (2022), 230–265, arXiv: 2110.15876 | DOI

[46] Song J., Qu C., Qiao Z., “A new integrable two-component system with cubic nonlinearity”, J. Math. Phys., 52 (2011), 013503, 9 pp. | DOI

[47] Tang H., Liu Z., “Well-posedness of the modified Camassa–Holm equation in Besov spaces”, Z. Angew. Math. Phys., 66 (2015), 1559–1580 | DOI

[48] Thirring W.E., “A soluble relativistic field theory”, Ann. Physics, 3 (1958), 91–112 | DOI

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

[50] Xia B., Zhou R., Qiao Z., “Darboux transformation and multi-soliton solutions of the Camassa–Holm equation and modified Camassa–Holm equation”, J. Math. Phys., 57 (2016), 103502, 12 pp., arXiv: 1506.08639 | DOI

[51] Xu J., Fan E., Long-time asyptotics behavior for the integrable modified Camassa–Holm equation with cubic nonlinearity, arXiv: 1911.12554

[52] Yang Y., Fan E., “On the long-time asymptotics of the modified Camassa–Holm equation in space-time solitonic regions”, Adv. Math., 402 (2022), 108340, 78 pp. | DOI

[53] Yu G.-F., Xu Z.-W., “Dynamics of a differential-difference integrable $(2+1)$-dimensional system”, Phys. Rev. E, 91 (2015), 062902, 10 pp. | DOI