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Existence of periodic wave of a BBM equation with delayed convection and weak diffusion

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Abstract

This paper focuses on the existence and uniqueness of periodic waves for a BBM equation with local strong generic delay convection and weak diffusion. We reduce the singular perturbed system to a regular perturbed system by constructing a locally invariant manifold according to geometric singular perturbation theory. The existence and uniqueness of periodic wave are given with sufficiently small perturbation parameter. Chebyshev criteria are applied to investigate the ratio of Abelian integrals. Moreover, the upper and lower bounds of the limiting wave speed are given.

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Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Sarkar, T., Roy, S., Raut, S., Mali, P.C.: Studies on the dust acoustic shock, solitary, and periodic waves in an unmagnetized viscous dusty plasma with two-temperature ions. Brazilian J. Phys. 53, 21 (2023)

    Google Scholar 

  2. Chadha, N.M., Tomar, S., Raut, S.: Parametric analysis of dust ion acoustic waves in superthermal plasmas through non-autonomous KdV framework. Commun. Non. Sci. Numer. Simul. 123, 107269 (2023)

    MathSciNet  MATH  Google Scholar 

  3. Wei, M.: Existence of kink waves to perturbed dispersive K(3,1) equation. J. Appl. Anal. Comput. 12, 712–719 (2022)

    MathSciNet  Google Scholar 

  4. Fan, F., Chen, X.: Dynamical behavior of traveling waves in a generalized VP-mVP equation with non-homogeneous power law nonlinearity. AIMS Math. 8, 17514–17538 (2023)

    MathSciNet  Google Scholar 

  5. Li, J.B., Chen, G., Zhou, Y.: Bifurcations and exact traveling wave solutions of two shallow water two-component systems. Int. J. Bifurcat. Chaos 31, 2150001 (2021)

    MathSciNet  MATH  Google Scholar 

  6. Ablowitz, M.J., Javier, V.: Solutions to the time dependent Schr\(\ddot{o}\)dinger and the Kadomtsev-Petviashvili equations. Phys. Rev. Lett. 78, 570–573 (1997)

    MathSciNet  Google Scholar 

  7. Wazwaz, A.M.: A reliable treatment of the physical structure for the nonlinear equation K(m, n). Appl. Math. Comput. 163, 1081–1095 (2005)

    MathSciNet  MATH  Google Scholar 

  8. Wazwaz, A.M.: Solutions of compact and noncompact structures for nonlinear Klein-Gordon-type equation. Appl. Math. Comput. 134, 487–500 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Ma, W.: Darboux transformations for a Lax integrable system in 2n dimensions. Lett. Math. Phys. 39, 33–49 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer-Verlag, New York (1993)

    MATH  Google Scholar 

  11. Li, J.B.: Singular Nonlinear Traveling Wave Equations: Bifurcation and Exact Solutions. Science Press, Beijing (2013)

    Google Scholar 

  12. Raut, S., Barman, R., Sarkar, T.: Integrability, breather, lump and quasi-periodic waves of non-autonomous Kadomtsev-Petviashvili equation based on Bell-polynomial approach. Wave Motion 119, 103125 (2023)

    MathSciNet  Google Scholar 

  13. Roy, S., Raut, S., Kairi, R.R., Chatterjee, P.: Integrability and the multi-soliton interactions of non-autonomous ZakharovCKuznetsov equation. Eur. Phys. J. Plus 137, 579 (2022)

    Google Scholar 

  14. Roy, S., Raut, S., Kairi, R.R., Chatterjee, P.: Bilinear B\(\ddot{\rm u }\)cklund, Lax pairs, breather waves, lump waves and soliton interaction of (2+1)-dimensional non-autonomous Kadomtsev-Petviashvili equation. Non. Dyn. 111, 5721–5741 (2023)

    Google Scholar 

  15. Korteweg, D., de Vries, G.: On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves. Philos. Mag. 39, 422–443 (1895)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  17. Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid. Mech. 78, 237–246 (1976)

    MATH  Google Scholar 

  18. Bateman, H.: Some recent researches on the motion of fluids. Mon. Wea. Rev 43, 163–170 (1915)

    Google Scholar 

  19. Camassa, R., Holm, D.: An integrable shallow water equation with peaked soliton. Phys. Rev. Lett. 71, 1661–1664 (1993)

    MathSciNet  MATH  Google Scholar 

  20. Benjamin, T.B., Bona, J.L., Mahony, J.J.: Model equations for long waves in nonlinear dispersive systems. Trans. Roy. Soc. Ser. A 272, 47–78 (1992)

    MathSciNet  MATH  Google Scholar 

  21. Bona, J.: On Solitary Waves and their Role in the Evolution of Long Waves, Applications of Nonlinear Analysis in the Physical Sciences. Pitman Press, Boston (1981)

    MATH  Google Scholar 

  22. Micu, S.: On the controllability of the linearized Benjamin-Bona-Mahony equation. SIAM J. Control. Optim. 39, 1677–1696 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Gupta, A.K., Hazarika, J.: On the solitary wave solutions of modified Benjamin-Bona-Mahony equation for unidirectional propagation of long waves. Pramana J. Phys. 94, 134 (2020)

    Google Scholar 

  24. Zhao, X., Xu, W., Li, S., Shen, J.: Bifurcations of traveling wave solutions for a class of the generalized Benjamin-Bona-Mahony equation Appl. Math. Comput. 175, 1760–1774 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Biswas, A.: 1-Soliton solution of Benjamin-Bona-Mahoney equation with dual-power law nonlinearity. Comm. Non. Sci. Numer. Simulat. 15, 2744–2746 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Singh, K., Gupta, R.K., Kumar, S.: Benjamin-Bona-Mahony (BBM) equation with variable coefficients: Similarity reductions and Painlev analysis. Appl. Math. Comput. 217, 7021–7027 (2011)

    MathSciNet  MATH  Google Scholar 

  27. Wazwaz, A.M.: Exact solution with compact and non-compact structures for the one-dimensional generalized Benjamin-Bona-Mahonyequation. Comm. Non. Sci. Numer. Simulat. 10, 855–867 (2005)

    MATH  Google Scholar 

  28. Chen, A., Guo, L., Deng, X.: Existence of solitary waves and periodic waves for a perturbed generalized BBM equation. J. Differ. Equat. 261, 5324–5349 (2016)

    MathSciNet  MATH  Google Scholar 

  29. Zhu, K., Wu, Y., Yu, Z., Shen, J.: New solitary wave solutions in a perturbed generalized BBM equation. Non. Dyn. 97, 2413–2423 (2019)

    MATH  Google Scholar 

  30. Zhang, L., Wang, J., Shchepakina, E., Sobolev, V.: New type of solitary wave solution with coexisting crest and trough for a perturbed wave equation. Non. Dyn. 106, 3479–3493 (2021)

    Google Scholar 

  31. Sun, X., Yu, P.: Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms. Dyn. Syst. Ser. B 24, 965–987 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Guo, L., Zhao, Y.: Existence of periodic waves for a perturbed quintic BBM equation. Dyn. Syst. Ser. B 40, 4689–4703 (2020)

    MathSciNet  MATH  Google Scholar 

  33. Dai, Y., Wei, M.: Existence and uniqueness of periodic waves for a perturbed sixtic generalized BBM equation. J. Appl. Anal. Comput. 1, 1–24 (2023)

    Google Scholar 

  34. Dai, Y., Wei, M.: Existence of periodic waves in a perturbed generalized BBM equation. Int. J. Bifur. Chaos 33, 2350060 (2023)

    MathSciNet  Google Scholar 

  35. Ogawa, T.: Travelling wave solutions to a perturbed Korteweg-de Vries equation. Hiroshima J. Math. 24, 401–422 (1994)

    MathSciNet  MATH  Google Scholar 

  36. Yan, W., Liu, Z., Liang, Y.: Existence of solitary waves and periodic waves to a perturbed generalized KdV equation. Math. Model. Anal. 19, 537–555 (2014)

    MathSciNet  MATH  Google Scholar 

  37. Chen, A., Guo, L., Huang, W.: Existence of kink waves and periodic waves for a perturbed defocusing mKdV equation. Qual. Theory Dyn. Syst. 17, 495–517 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Sun, X., Huang, W., Cai, J.: Coexistence of the solitary and periodic waves in convecting shallow water fluid. Non. Anal. Real World Appl. 53, 103067 (2020)

    MathSciNet  MATH  Google Scholar 

  39. Wei, M., He, L.: Existence of periodic wave for a perturbed MEW equation. AIMS Math. 8, 11557–11571 (2023)

    MathSciNet  Google Scholar 

  40. Du, Z., Li, J., Li, X.: The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach. J. Funct. Ana. 275, 988–1007 (2018)

    MathSciNet  MATH  Google Scholar 

  41. Zhuang, K., Du, Z., Lin, X.: Solitary waves solutions of singularly perturbed higher-order KdV equation via geometric singular perturbation method. Non. Dyn. 80, 629–635 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Du, Z., Li, J.: Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation. J. Differ. Equat. 306, 418–438 (2022)

    MathSciNet  MATH  Google Scholar 

  43. Wang, J., Zhang, L., Li, J.: New solitary wave solutions of a generalized BBM equation with distributed delays. Non. Dyn. 111(5), 4631–4643 (2023)

    Google Scholar 

  44. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equat. 31, 53–98 (1979)

    MathSciNet  MATH  Google Scholar 

  45. Jones, C.K.R.T.: Geometric singular perturbation theory. Lecture Notes Math. 1609, 45–118 (1994)

    Google Scholar 

  46. Kuehn, C.: Multiple Time Scale Dynamics. Springer, New York (2014)

    MATH  Google Scholar 

  47. Grau, M., Mañsas, F., Villadelprat, J.: A Chebyshev criterion for Abelian integrals. Trans. Am. Math. Soc. 363, 109–129 (2011)

    MathSciNet  MATH  Google Scholar 

  48. Manosas, F., Villadelprat, J.: Bounding the number of zeros of certain Abelian integrals. J. Differ. Equ. 251, 1656–1669 (2011)

    MathSciNet  MATH  Google Scholar 

  49. Han, M., Yu, P.: Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles. Springer, New York (2012)

    MATH  Google Scholar 

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Funding

This research was supported by the 681 National Natural Science Foundation of China (2001121).

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Authors and Affiliations

  1. School of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China

    Minzhi Wei

  2. School of Mathematics and Quantitative Economics, Guangxi University of Finance and Economics, Nanning, 530003, Guangxi, People’s Republic of China

    Minzhi Wei

  3. China-ASEAN Institute of Statistics, Guangxi University of Finance and Economics, Nanning, 530003, Guangxi, People’s Republic of China

    Liping He

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  1. Minzhi Wei
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Wei, M., He, L. Existence of periodic wave of a BBM equation with delayed convection and weak diffusion. Nonlinear Dyn 111, 17413–17425 (2023). https://doi.org/10.1007/s11071-023-08743-w

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