AIMS Mathematics, 2019, 4(5): 1450-1465. doi: 10.3934/math.2019.5.1450

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On traveling wave solutions of a class of KdV-Burgers-Kuramoto type equations

School of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China

In the paper, the traveling wave solutions of a KdV–Burgers-Kuramoto type equation with arbitrary power nonlinearity are considered. Lie symmetry analysis method on the equation is performed, which shows that the equation possesses traveling wave solutions. By qualitative analysing the equivalent autonomous system of the traveling wave equation of the equation, the existence of the traveling wave solutions of the equation is presented. Through analysing the associated determining system, the non-trivial infinitesimal generator of Lie symmetry admitted by the traveling wave solutions equation under the certain parametric conditions is found. The traveling wave solutions of the KdV–Burgers-Kuramoto type equation by solving the invariant surface condition equation under the certain parametric conditions are obtained.
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References

1. M. R. Miura, Backlund Transfortion, New York: Springer-Verlag, 1978.

2. W. Peng, S. Tian, T. Zhang, Dynamics of breather waves and higher-order rogue waves in a coupled nonlinear Schrödinger equation, Europhysics Letters, 123 (2018), 50005.

3. X. Wang, T. Zhang, M. Dong, Dynamics of the breathers and rogue waves in the higher-order nonlinear Schrödinger equation, Appl. Math. Lett., 86 (2018), 298304.

4. L. Feng, T. Zhang, Breather wave, rogue wave and solitary wave solutions of a coupled nonlinear Schrödinger equation, Appl. Math. Lett., 78 (2018), 133-140.

5. D. Guo, S. Tian, T. Zhang, Integrability, soliton solutions and modulation instability analysis of a (2+1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation, Comput. Math.Appl., 77 (2019), 770-778.    

6. L. Feng, S. Tian, T. Zhang, Solitary wave, breather wave and rogue wave solutions of an inhomogeneous fifth-order nonlinear Schrödinger equation from Heisenberg ferromagnetism, Rocky MT J. Math., 49 (2019), 29-45.

7. W. Peng, S. Tian, T. Zhang, Breather waves and rational solutions in the (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Comput. Math. Appl., 77 (2019), 715-723.    

8. M. Dong, S. F. Tian, X. W. Yan, et al, Solitary waves, homoclinic breather waves and rogue waves of the (3+1)-dimensional Hirota bilinear equation, Comput. Math. Appl., 75 (2018), 957-964.    

9. X. Yan, S. Tian, M. Dong, et al. Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation, Comput. Math. Appl., 76 (2018), 179-186.

10. S. Tian, Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via theFokas method, J. Differ. Equations, 262 (2017), 506-558.    

11. W. Ma, T. Huang, Y. Zhang, A multiple exp-function method for nonlinear differential equations and its application, Phys. Scripta, 82 (2010), 065003.

12. J. He, X. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals, 30 (2006), 700-708.    

13. A. M. Wazwaz, Two reliable methods for solving variants of the KdV equation with compact and noncompact structures, Chaos, Solitons and Fractals, 28 (2006), 454-462.    

14. T. Xia, B. Li, H. Zhang, New explicit and exact solutions for the Nizhnik-Novikov-Vesselov equation, Applied Mathematics E-Notes, 1 (2001), 139-142.

15. Z. Feng, The first integral method to study the Burgers-Korteweg-de Vries equation, Journal of Physics A, 35 (2002), 343-349.    

16. Z. Feng, X. Wang, The first integral method to the two-dimensional Burgers-KdV equation, Phys. Lett. A, 308 (2002), 173-178.

17. W. Ma, J. H. Lee, A transfortiom rational function method and exact solutions to (3+1)-dimensional Jimbo-Miwa equation, Chaos, Solitons and Fractals, 42 (2009), 1356-1363.    

18. P. J. Olver, Applications of Lie groups to differential equations, New York: Springer-Verlag, 1999.

19. W. G. Bluman, C. S. Anco, Symmetry and integration methods for differential equations, New York: Springer-Verlag, 2002.

20. X. Wang, S. Tian and T. Zhang, Characteristics of the breather and rogue waves in a (2+1)-dimensional nonlinear Schrödinger equation, proceedings of the american mathematical society, 146 (2018), 3353-3365.    

21. S. Tian, T. Zhang, Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition, proceedings of the american mathematical society, 146 (2018), 1713-1729.

22. J. Li, Singular Traveling Wave Equations: Bifurcations and Exact Solutions, Beijing: Science Press, 2013.

23. X. Wang, S. Tian, C. Qin, et al, A Lie symmetry analysis, conservation laws and exact solutions of the generalized time fractional Burgers equation, Europhysics Letters, 114 (2016), 20003.

24. Z. Feng, Traveling waves to a reaction-diffusion equation, Discret. Contin. Dyn. S., Supp., 12 (2007), 382-390.

25. H. Liu, J. Li, L. Liu, Lie symmetry analysia, optimal systems and exact solutions to the fifth-order KdV types of equations, J. Math. Anal. Appl., 368 (2010), 551-558.    

26. H. Liu, J. Li, Q. Zhang, Lie symmetry analysis and exact explicit solutions for general Burgers' equation, J. Comput. Appl. Math., 228 (2009), 1-9.    

27. M. L. Gandarias, C. M. Khalique, Symmetries, solutions and conservation laws of a class of nonlinear dispersive wave equations, Commun. Nonlinear Sci., 32 (2016), 114-121.    

28. Y. Hu, C. Xue, One-parameter Lie groups and inverse integrating factors of n-th order autonomous systems, J. Math. Anal. Appl., 388 (2012), 617-626.    

29. Y. Hu, K. Guan, Techniques for searching first integrals by Lie group and application to gyroscope system, Sci. China Math., 48 (2005), 1135-1143.    

30. Y. Kuramoto, T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermel equilibrium, Prog. Theor. Phys., 55 (1976), 356-369.    

31. J. Topper, T. Kawahara, Approximate equation for long nonlinear waves on a viscous fluid, J. Phys. Soc. Jpn, 44 (1978), 663-666.    

32. V. Y. Shkadov, Solitary waves in a layer of viscous liquid, Fluid Dynamics, 12 (1977), 52-55.    

33. B. I. Cohen, J. A. Krommers, W. M. Tang, et al. Non-linear saturation of the dissipative trapped-ion mode by mode coupling, Nucl. Fusion, 16 (1976), 971-992.    

34. S. D. Liu, S. K. Liu, Z. Huang, et al. On a class of nonlinear Schrödinger equation III, Progress in Natural Science, 9 (1999), 912-918.

35. Z. Fu, S. D. Liu, S. K. Liu, New exact solutions to the Kdv-Burgers-Kuramoto equation, Chaos, Solitons and Fractals, 23 (2005), 609-616.    

36. Abdul-Majid Wazwaz, Partial differential equations and solitary waves theory, Beijing: Higher Education Press, 2009.

37. X. Chen, Z. Fu, S. Liu, Periodic solutions to KdV-Burgers-Kuramoto equations, Commun. Theor. Phys., 45 (2006), 815-818.    

38. Y. Fu, Z. Liu, Persistence of travelling fronts of Kdv-Burgers-Kuramoto equation, Applied Mathematics and Computation, Chaos, Solitons and Fractals, 216 (2010), 2199-2206.

39. J. Nickel, Travelling wave solutions of the Kuramoto-Sivashinsky equation, Chaos, Solitons and Fractals, 33 (2007), 1376-1382.    

40. Y. Hu, Lie symmetry analysis and exact solutions to a class of new KdV-Burgers-Kuramoto type equation, Chinese Control and Decision Conference, (2016), 6705-6709.

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