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Nonlinear Dynamics

, Volume 95, Issue 1, pp 273–291 | Cite as

Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq–Burgers equation

  • Min-Jie Dong
  • Shou-Fu TianEmail author
  • Xue-Wei Yan
  • Tian-Tian Zhang
Original Paper
  • 271 Downloads

Abstract

We consider the classical Boussinesq–Burgers (BB) equation, which describes the propagation of shallow water waves. Based on the truncated painlevé expansion method and consistent Riccati expansion method, we successfully obtain its nonlocal symmetry and Bäcklund transformation. By introducing auxiliary-dependent variables for the nonlocal symmetry, we find the corresponding Lie point symmetries. By considering the consistent tanh expansion method, the interaction solution of soliton–cnoidal wave for the classical BB equation is studied by using the Jacobi elliptic function. The multi-solitary wave solutions are also obtained by introducing a linear combination of N exponential functions. Moreover, the conservation laws of the equation are successfully obtained with a detailed derivation.

Keywords

The classical Boussinesq–Burgers equation Truncated painlevé expansion Interaction solutions Lie point symmetry Conservation laws 

Notes

Acknowledgements

We express our sincere thanks to the Editor and Reviewers for their valuable comments. This work was supported by the Jiangsu Province Natural Science Foundation of China under Grant No. BK20181351, the Postgraduate Research and Practice Program of Education and Teaching Reform of CUMT under Grant No. YJSJG_2018_036, the “Qinglan Engineering project” of Jiangsu Universities, the National Natural Science Foundation of China under Grant No. 11301527, the Fundamental Research Fund for the Central Universities under the Grant No. 2017XKQY101, and the General Financial Grant from the China Postdoctoral Science Foundation under Grant Nos. 2015M57 0498 and 2017T100413.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bluman, G.W., Kumei, S.: Symmetries and Differential Equations, Graduate Texts in Math, 81. Springer, New York (1989)zbMATHCrossRefGoogle Scholar
  3. 3.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1999)Google Scholar
  4. 4.
    Ibragimov, N.H. (ed.): CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1. CRC Press, Boca Raton (1994)zbMATHGoogle Scholar
  5. 5.
    Lou, S.Y., Hu, X.B.: Non-local symmetries via Darboux transformations. J. Phys. A Math. Gen. 30, L95 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fan, E.G.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Wazwaz, A.M.: Partial Differential Equations: Methods and Applications. Balkema Publishers, Delft (2002)zbMATHGoogle Scholar
  8. 8.
    Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83, 591–596 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dai, C.Q., Liu, J., Fan, Y., Yu, D.G.: Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality. Nonlinear Dyn. 88(2), 1373–1383 (2017)CrossRefGoogle Scholar
  11. 11.
    Ding, D.J., Jin, D.Q., Dai, C.Q.: Analytical solutions of differential-difference sine-Gordon equation. Therm. Sci. 21, 1701–1705 (2017)CrossRefGoogle Scholar
  12. 12.
    Dai, C.Q., Wang, Y.Y., Fan, Y., Yu, D.G.: Reconstruction of stability for Gaussian spatial solitons in quintic–septimal nonlinear materials under PT-symmetric potentials. Nonlinear Dyn. 92, 1351–1358 (2018)CrossRefGoogle Scholar
  13. 13.
    Wang, Y.Y., Zhang, Y.P., Dai, C.Q.: Re-study on localized structures based on variable separation solutions from the modified tanh-function method. Nonlinear Dyn. 83, 1331–1339 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wang, Y.Y., Chen, L., Dai, C.Q., Zheng, J., Fan, Y.Y.: Exact vector multipole and vortex solitons in the media with spatially modulated cubic–quintic nonlinearity. Nonlinear Dyn. 90, 1269–1275 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Wang, D.S., Wang, X.L.: Long-time asymptotics and the bright N-soliton solutions of the Kundu–Eckhaus equation via the Riemann–Hilbert approach. Nonlinear Anal. 41, 334–361 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Lou, S.Y., Hu, X., Chen, Y.: Nonlocal symmetries related to Bäcklund transformation and their applications. J. Phys. A Math. Theor. 45(15), 155209 (2012)zbMATHCrossRefGoogle Scholar
  17. 17.
    Lou, S.Y.: Consistent Riccati Expansion for Integrable Systems. Stud. Appl. Math. 134(3), 372–402 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Lou, S.Y., Hu, H.C., Tang, X.Y.: Interactions among periodic waves and solitary waves of the (N + 1)-dimensional sine-Gordon field. Phys. Rev. E 71, 036604 (2005)CrossRefGoogle Scholar
  19. 19.
    Chen, C.L., Lou, S.Y.: CTE solvability, nonlocal symmetries and exact solutions of dispersive water wave system. Commun. Theor. Phys. 61, 545–550 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hu, X.R., Lou, S.Y., Chen, Y.: Explicit solutions from eigenfunction symmetry of the Korteweg–de Vries equation. Phys. Rev. E 85, 056607 (2012)CrossRefGoogle Scholar
  21. 21.
    Ren, B., Lin, J.: Interaction behaviours between soliton and cnoidal periodic waves for the cubic generalised Kadomtsev–Petviashvili equation. Z. Naturforsch. 70a, 539 (2015)Google Scholar
  22. 22.
    Cheng, W.G., Li, B., Chen, Y.: Construction of Soliton–Cnoidal wave interaction solution for the (2 + 1)-dimensional breaking soliton equation. Commun. Theor. Phys. 63, 549 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Hu, H.C., Hu, X., Feng, B.F.: Nonlocal symmetry and consistent tanh expansion method for the coupled integrable dispersionless equation. Z. Naturforsch. 71, 235 (2016)Google Scholar
  24. 24.
    Xin, X.P., Liu, X.Q.: Interaction solutions for (1 + 1)-dimensional higher-order Broer–Kaup system. Commun. Theor. Phys. 66, 479 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Chen, J.C., Chen, Y.: Nonlocal symmetry constraints and exact interaction solutions of the (2 + 1)-dimensional modified generalized long dispersive wave equation. J. Nonlinear Math. Phys. 21, 454 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Yan, X.W., Tian, S.F., Dong, M.J., Wang, X.B., Zhang, T.T.: Nonlocal symmetries, conservation laws and interaction solutions of the generalised dispersive modified Benjamin–Bona–Mahony equation. Z. Naturforsch. A 73(5), 399–405 (2018)Google Scholar
  27. 27.
    Tian, S.F., Zhang, Y.F., Feng, B.L., Zhang, H.Q.: On the Lie algebras, generalized symmetries and Darboux transformations of the fifth-order evolution equations in shallow water. Chin. Ann. Math. B. 36(4), 543–560 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Feng, L.L., Tian, S.F., Zhang, T.T., Zhou, J.: Nonlocal symmetries, consistent Riccati expansion, and analytical solutions of the variant Boussinesq system. Z. Naturforsch. A 72(7), 655–663 (2017)CrossRefGoogle Scholar
  29. 29.
    Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Lie symmetry analysis, analytical solutions, and conservation laws of the generalised Whitham–Broer–Kaup–Like equations. Z. Naturforsch. A 72(3), 269–279 (2017)CrossRefGoogle Scholar
  30. 30.
    Feng, L.L., Tian, S.F., Zhang, T.T.: Nonlocal symmetries and consistent Riccati expansions of the (2 + 1)-dimensional dispersive long wave equation. Z. Naturforsch. A 72(5), 425–431 (2017)CrossRefGoogle Scholar
  31. 31.
    Noether, E.: Invariante variations probleme. Nachr. Ges. Wiss. Göttingen 1918, 235 (1918)zbMATHGoogle Scholar
  32. 32.
    Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333(1), 311–328 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Geng, X.G., Wu, Y.T.: Finite-band solutions of the classical Boussinesq–Burgers equations. J. Math. Phys. 40(6), 2971–2982 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Rui, X.: Darboux transformations and soliton solutions for classical Boussinesq–Burgers equation. Commun. Theor. Phys. 50, 579–582 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Rady, A.S.A., Osman, E.S., Khalfallah, Mo: Multi-soliton solution, rational solution of the Boussinesq–Burgers equations. Commun. Nonlinear Sci. Numer. Simul. 15(5), 1172–1176 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Wang, Y.H.: CTE method to the interaction solutions of Boussinesq–Burgers equations. Appl. Math. Lett. 38(38), 100–105 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Wang, P., Tian, B., Liu, W.J., Xing, L., Jiang, Y.: Lax pair, Bäcklund transformation and multi-soliton solutions for the Boussinesq–Burgers equations from shallow water waves. Appl. Math. Comput. 218(5), 1726–1734 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Wazwaza, A.M.: A variety of soliton solutions for the Boussinesq–Burgers equation and the higher-order Boussinesq–Burgers equation. Filomat 31(3), 831–840 (2017)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Ren, B.: Interaction solutions for mKP equation with nonlocal symmetry reductions and CTE method. Phys. Scr. 90, 065206 (2015)CrossRefGoogle Scholar
  40. 40.
    Keane, A.J., Mushtaq, A., Wheatland, M.S.: Alfven solitons in a Fermionic quantum plasma. Phys. Rev. E 83, 066407 (2011)CrossRefGoogle Scholar
  41. 41.
    Yan, X.W., Tian, S.F., Dong, M.J., Zhou, L., Zhang, T.T.: Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2 + 1)-dimensional generalized breaking soliton equation. Comput. Math. Appl. 76(1), 179–186 (2018)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Wazwaz, A.M.: Multiple-soliton solutions for the fifth-order Caudrey–Dodd–Gibbon equation. Appl. Math. Comput. 197, 719–724 (2008)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Wazwaz, A.M., El-Tantawy, S.A.: Solving the (3 + 1)-dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method. Nonlinear. Dyn 88, 3017–3021 (2017)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Tian, S.F.: Asymptotic behavior of a weakly dissipative modified two-component Dullin–Gottwald–Holm system. Appl. Math. Lett. 83, 65–72 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Qin, C.Y., Tian, S.F., Wang, X.B., Zhang, T.T., Li, J.: Rogue waves, bright-dark solitons and traveling wave solutions of the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation. Comput. Math. Appl. 75(12), 4221–4231 (2018)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Wang, X.B., Tian, S.F., Zhang, T.T.: Characteristics of the breather and rogue waves in a (2 + 1)-dimensional nonlinear Schrödinger equation. Proc. Am. Math. Soc. 146(8), 3353–3365 (2018)zbMATHCrossRefGoogle Scholar
  47. 47.
    Yan, X.W., Tian, S.F., Dong, M.J., Zou, L.: Bäcklund transformation, rogue wave solutions and interaction phenomena for a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Nonlinear Dyn. 92(2), 709–720 (2018)zbMATHCrossRefGoogle Scholar
  48. 48.
    Dong, M.J., Tian, S.F., Yan, X.W., Zou, L.: Solitary waves, homoclinic breather waves and rogue waves of the (3 + 1)-dimensional Hirota bilinear equation. Comput. Math. Appl. 75(3), 957–964 (2018)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Wang, X.B., Tian, S.F., Feng, L.L., Zhang, T.T.: On quasi-periodic waves and rogue waves to the (4 + 1)-dimensional nonlinear Fokas equation. J. Math. Phys. 59, 073505 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Feng, L.L., Tian, S.F., Wang, X.B., Zhang, T.T.: Rogue waves, homoclinic breather waves and soliton waves for the (2 + 1)-dimensional B-type Kadomtsev–Petviashvili equation. Appl. Math. Lett. 65, 90–97 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Wang, X.B., Tian, S.F., Yan, H., Zhang, T.T.: On the solitary waves, breather waves and rogue waves to a generalized (3 + 1)-dimensional Kadomtsev–Petviashvili equation. Comput. Math. Appl. 74, 556–563 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Feng, L.L., Zhang, T.T.: Breather wave, rogue wave and solitary wave solutions of a coupled nonlinear Schrödinger equation. Appl. Math. Lett. 78, 133–140 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Dynamics of the breathers, rogue waves and solitary waves in the (2 + 1)-dimensional Ito equation. Appl. Math. Lett. 68, 40–47 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Characteristics of the solitary waves and rogue waves with interaction phenomena in a generalized (3 + 1)-dimensional Kadomtsev–Petviashvili equation. Appl. Math. Lett. 72, 58–64 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Wang, X.B., Zhang, T.T., Dong, M.J.: Dynamics of the breathers and rogue waves in the higher-order nonlinear Schrödinger equation. Appl. Math. Lett. 86, 298–304 (2018)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Tu, J.M., Tian, S.F., Xu, M.J., Ma, P.L., Zhang, T.T.: On periodic wave solutions with asymptotic behaviors to a (3 + 1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics. Comput. Math. Appl. 72, 2486–2504 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Qin, C.Y., Tian, S.F., Zou, L., Ma, W.X.: Solitary wave and quasi–periodic wave solutions to a (3 + 1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation. Adv. Appl. Math. Mech. 10(4), 948–977 (2018)MathSciNetGoogle Scholar
  58. 58.
    Tian, S.F.: The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method. Proc. R. Soc. Lond. A 472, 20160588 (2016)zbMATHCrossRefGoogle Scholar
  59. 59.
    Tian, S.F.: Initial-boundary value problems for the coupled modified Korteweg–de Vries equation on the interval. Commun. Pure Appl. Anal. 173, 923–957 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Tian, S.F., Zhang, T.T.: Long-time asymptotic behavior for the Gerdjikov–Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition. Proc. Am. Math. Soc. 146(4), 1713–1729 (2018)zbMATHCrossRefGoogle Scholar
  61. 61.
    Tian, S.F.: Initial-boundary value problems of the coupled modified Korteweg–de Vries equation on the half-line via the Fokas method. J. Phys. A Math. Theor. 50, 395204 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Tian, S.F.: Initial-boundary value problems for the general coupled nonlinear Schrödinger equations on the interval via the Fokas method. J. Differ. Equ. 262, 506–558 (2017)zbMATHCrossRefGoogle Scholar
  63. 63.
    Xu, M.J., Tian, S.F., Tu, J.M., Zhang, T.T.: Bäcklund transformation, infinite conservation laws and periodic wave solutions to a generalized (2 + 1)-dimensional Boussinesq equation. Nonlinear Anal. Real World Appl. 31, 388–408 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Tong, S., Sun, K., Sui, S.: Observer-based adaptive fuzzy decentralized optimal control design for strict-feedback nonlinear large-scale systems. IEEE Trans. Fuzzy Syst. 99, 1–1 (2018)Google Scholar
  65. 65.
    Sun, K., Sui, S., Tong, S.: Fuzzy adaptive decentralized optimal control for strict feedback nonlinear large-scale systems. IEEE Trans. Cybern. 48(4), 1326–1339 (2018)CrossRefGoogle Scholar
  66. 66.
    Sun, K., Li, Y., Tong, S.: Fuzzy adaptive output feedback optimal control design for strict-feedback nonlinear systems. IEEE Trans. Cybern. 99, 1–12 (2017)CrossRefGoogle Scholar
  67. 67.
    Sun, K., Sui, S., Tong, S.: Optimal adaptive fuzzy FTC design for strict-feedback nonlinear uncertain systems with actuator faults. Fuzzy Sets Syst. 316, 20–34 (2016)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and Institute of Mathematical PhysicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China

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