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

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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.

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References

  1. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  2. Bluman, G.W., Kumei, S.: Symmetries and Differential Equations, Graduate Texts in Math, 81. Springer, New York (1989)

    Book  MATH  Google Scholar 

  3. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1999)

    Google Scholar 

  4. Ibragimov, N.H. (ed.): CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1. CRC Press, Boca Raton (1994)

    MATH  Google Scholar 

  5. Lou, S.Y., Hu, X.B.: Non-local symmetries via Darboux transformations. J. Phys. A Math. Gen. 30, L95 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fan, E.G.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wazwaz, A.M.: Partial Differential Equations: Methods and Applications. Balkema Publishers, Delft (2002)

    MATH  Google Scholar 

  8. Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83, 591–596 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  11. Ding, D.J., Jin, D.Q., Dai, C.Q.: Analytical solutions of differential-difference sine-Gordon equation. Therm. Sci. 21, 1701–1705 (2017)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  17. Lou, S.Y.: Consistent Riccati Expansion for Integrable Systems. Stud. Appl. Math. 134(3), 372–402 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Xin, X.P., Liu, X.Q.: Interaction solutions for (1 + 1)-dimensional higher-order Broer–Kaup system. Commun. Theor. Phys. 66, 479 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  31. Noether, E.: Invariante variations probleme. Nachr. Ges. Wiss. Göttingen 1918, 235 (1918)

    MATH  Google Scholar 

  32. Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333(1), 311–328 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Geng, X.G., Wu, Y.T.: Finite-band solutions of the classical Boussinesq–Burgers equations. J. Math. Phys. 40(6), 2971–2982 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rui, X.: Darboux transformations and soliton solutions for classical Boussinesq–Burgers equation. Commun. Theor. Phys. 50, 579–582 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, Y.H.: CTE method to the interaction solutions of Boussinesq–Burgers equations. Appl. Math. Lett. 38(38), 100–105 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  39. Ren, B.: Interaction solutions for mKP equation with nonlocal symmetry reductions and CTE method. Phys. Scr. 90, 065206 (2015)

    Article  Google Scholar 

  40. Keane, A.J., Mushtaq, A., Wheatland, M.S.: Alfven solitons in a Fermionic quantum plasma. Phys. Rev. E 83, 066407 (2011)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  42. Wazwaz, A.M.: Multiple-soliton solutions for the fifth-order Caudrey–Dodd–Gibbon equation. Appl. Math. Comput. 197, 719–724 (2008)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  44. Tian, S.F.: Asymptotic behavior of a weakly dissipative modified two-component Dullin–Gottwald–Holm system. Appl. Math. Lett. 83, 65–72 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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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.

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  1. School of Mathematics and Institute of Mathematical Physics, China University of Mining and Technology, Xuzhou, 221116, People’s Republic of China

    Min-Jie Dong, Shou-Fu Tian, Xue-Wei Yan & Tian-Tian Zhang

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  1. Min-Jie Dong
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Dong, MJ., Tian, SF., Yan, XW. et al. Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq–Burgers equation. Nonlinear Dyn 95, 273–291 (2019). https://doi.org/10.1007/s11071-018-4563-9

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