Abstract
The integrability and multi-shock wave solutions of the DJKM equation are studied by means of Bell polynomials scheme, Hirota bilinear method, and symbolic computation. A more generalized bilinear system of the DJKM equation is constructed via Bell polynomials scheme. Moreover, Lax pair and infinite conservation laws of this equation are first obtained via its corresponding Bell-polynomials-type Bäcklund transformation. Furthermore, the multi-shock wave solutions are also obtained by applying standard Hirota bilinear method, and the propagation and collision of shock waves are graphically demonstrated by graphs.
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References
Hu, X.B., Li, Y.: Bäcklund transformation and nonlinear superposition formula of DJKM equation. Acta Math. Sci. 11, 164–172 (1991). (in Chinese)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Jimbo, M., Wiwa, T. (eds.) Proceeding of the RIMS Symposium on Nonlinear Integrable Systems-Classical and Quantum Theoroy, World Scientific, Singapore (1983)
Dorizzi, B., Grammaticos, B., Ramani, A., Winternitz, P.: Are all the equations of the Kadomtsev–Petviashvili hierarchy integrable? J. Math. Phys. 27, 2848–2852 (1986)
Bell, E.T.: Exponential polynomials. Ann. Math. 35, 258–277 (1934)
Gilson, C., Lambert, F., Nimmo, J., Willox, R.: On the combinatorics of the Hirota \(D\)-operators. Proc. R. Soc. Lond. A 452, 223–234 (1996)
Lambert, F., Springael, J.: From soliton equations to their zero curvature formulation. Acta Appl. Math. 102, 147–178 (2008)
Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, New York (2004)
Liu, Q.P., Hu, X.B., Zhang, M.X.: Supersymmetric modified Korteweg–de Vries equation: bilinear approach. J. Phys. A 18, 1597–1603 (2005)
Hu, X.B., Li, C.X., Nimmo, J.J.C., Yu, G.F.: An integrable symmetric (2+1)-dimensional Lotka–Volterra equation and a family of its solutions. J. Phys. A 38, 195–204 (2005)
Hietarinta, J., Zhang, D.J.: Multisoliton solutions to the lattice Boussinesq equation. J. Math. Phys. 51, 033505 (2010)
Feng, B.F., Maruno, K.I., Ohta, Y.: On the \(\tau \)-functions of the Degasperis–Procesi equation. J. Phys. A 46, 045205 (2013)
Wazwaz, A.M., Zha, Q.L.: Nonsingular complexiton solutions for two higher-dimensional fifth-order nonlinear integrable equations. Phys. Scr. 88, 025001 (2013)
Dai, H.H., Fan, E.G., Geng, X.G.: Periodic wave solutions of nonlinear equations by Hirota’s bilinear method. arXiv:nlin/0602015
Fan, E.G., Hon, Y.C.: Quasiperiodic waves and asymptotic behavior for Bogoyavlenskii’s breaking soliton equation in (2+1) dimensions. Phys. Rev. E 78, 036607 (2008)
Ma, W.X., Zhou, R.G., Gao, L.: Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (2\(+\)1) dimensions. Mod. Phys. Lett. A 21, 1677–1688 (2009)
Zhang, Y., Song, Y., Cheng, L., Ge, J.Y., Wei, W.W.: Exact solutions and Painlevé analysis of a new (2+1)-dimensional generalized KdV equation. Nonlinear Dyn. 68, 445–458 (2012)
Fan, E.G.: The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials. Phys. Lett. A 375, 493–497 (2011)
Fan, E.G., Hon, Y.C.: Super extension of Bell polynomials with applications to supersymmetric equations. J. Math. Phys. 53, 013503 (2012)
Wang, Y.H., Chen, Y.: Integrability of the modified generalised Vakhnenko equation. J. Math. Phys. 53, 123504 (2012)
Wang, Y.H., Chen, Y.: Binary Bell polynomial manipulations on the integrability of a generalized (2+1)-dimensional Korteweg–de Vries equation. J. Math. Anal. Appl. 400, 624–634 (2013)
Luo, L.: New exact solutions and Bäcklund transformation for Boiti–Leon–Manna–Pempinelli equation. Phys. Lett. A 375, 1059–1063 (2011)
Wang, Y.F., Tian, B., Wang, P., Li, M., Jiang, Y.: Bell-polynomial approach and soliton solutions for the Zhiber–Shabat equation and (2+1)-dimensional Gardner equation with symbolic computation. Nonlinear Dyn. 69, 2031–2040 (2012)
Tian, S.F., Zhang, H.Q.: On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation. J. Phys. A 45, 055203 (2012)
Ma, W.X.: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys. 411, 012021 (2013)
Zhang, Y.F., Tam, H.: Discussion on integrable properties for higher-dimensional variable-coefficient nonlinear partial differential equations. J. Math. Phys. 54, 013516 (2013)
Wang, H., Xia, T.C.: Bell polynomial approach to an extended Korteweg–de Vries equation. Math. Method Appl. Sci. (2013). doi:10.1002/mma.2908
Miao, Q., Wang, Y.H., Chen, Y., Yang, Y.Q.: PDEBellII: a Maple package for finding bilinear forms, bilinear Bäcklund transformations, Lax pairs and conservation laws of the KdV-type equations. Comput. Phys. Commun. 185, 357–367 (2014)
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This work is supported by the National Natural Science Foundation of China under Grant No. 11071159.
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Wang, YH., Wang, H. & Temuer, C. Lax pair, conservation laws, and multi-shock wave solutions of the DJKM equation with Bell polynomials and symbolic computation. Nonlinear Dyn 78, 1101–1107 (2014). https://doi.org/10.1007/s11071-014-1499-6
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DOI: https://doi.org/10.1007/s11071-014-1499-6