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Lax pair, conservation laws, and multi-shock wave solutions of the DJKM equation with Bell polynomials and symbolic computation

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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

  1. Hu, X.B., Li, Y.: Bäcklund transformation and nonlinear superposition formula of DJKM equation. Acta Math. Sci. 11, 164–172 (1991). (in Chinese)

    Google Scholar 

  2. 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)

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bell, E.T.: Exponential polynomials. Ann. Math. 35, 258–277 (1934)

    Article  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lambert, F., Springael, J.: From soliton equations to their zero curvature formulation. Acta Appl. Math. 102, 147–178 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, New York (2004)

    Book  MATH  Google Scholar 

  8. Liu, Q.P., Hu, X.B., Zhang, M.X.: Supersymmetric modified Korteweg–de Vries equation: bilinear approach. J. Phys. A 18, 1597–1603 (2005)

    MathSciNet  MATH  Google Scholar 

  9. 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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hietarinta, J., Zhang, D.J.: Multisoliton solutions to the lattice Boussinesq equation. J. Math. Phys. 51, 033505 (2010)

    Article  MathSciNet  Google Scholar 

  11. Feng, B.F., Maruno, K.I., Ohta, Y.: On the \(\tau \)-functions of the Degasperis–Procesi equation. J. Phys. A 46, 045205 (2013)

    Article  MathSciNet  Google Scholar 

  12. Wazwaz, A.M., Zha, Q.L.: Nonsingular complexiton solutions for two higher-dimensional fifth-order nonlinear integrable equations. Phys. Scr. 88, 025001 (2013)

    Article  Google Scholar 

  13. Dai, H.H., Fan, E.G., Geng, X.G.: Periodic wave solutions of nonlinear equations by Hirota’s bilinear method. arXiv:nlin/0602015

  14. 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)

    Article  MathSciNet  Google Scholar 

  15. 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)

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fan, E.G.: The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials. Phys. Lett. A 375, 493–497 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fan, E.G., Hon, Y.C.: Super extension of Bell polynomials with applications to supersymmetric equations. J. Math. Phys. 53, 013503 (2012)

    Article  MathSciNet  Google Scholar 

  19. Wang, Y.H., Chen, Y.: Integrability of the modified generalised Vakhnenko equation. J. Math. Phys. 53, 123504 (2012)

    Article  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Luo, L.: New exact solutions and Bäcklund transformation for Boiti–Leon–Manna–Pempinelli equation. Phys. Lett. A 375, 1059–1063 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tian, S.F., Zhang, H.Q.: On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation. J. Phys. A 45, 055203 (2012)

    Article  MathSciNet  Google Scholar 

  24. Ma, W.X.: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys. 411, 012021 (2013)

    Google Scholar 

  25. Zhang, Y.F., Tam, H.: Discussion on integrable properties for higher-dimensional variable-coefficient nonlinear partial differential equations. J. Math. Phys. 54, 013516 (2013)

    Article  MathSciNet  Google Scholar 

  26. 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

  27. 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)

    Article  Google Scholar 

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Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant No. 11071159.

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Authors and Affiliations

  1. College of Art and Sciences, Shanghai Maritime University, Shanghai, 201306, People’s Republic of China

    Yun-Hu Wang, Hui Wang & Chaolu Temuer

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  1. Yun-Hu Wang
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  2. Hui Wang
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Correspondence to Yun-Hu Wang.

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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

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