August  2020, 40(8): 4689-4703. doi: 10.3934/dcds.2020198

Existence of periodic waves for a perturbed quintic BBM equation

School of Mathematics(Zhuhai), Sun Yat-Sen university, Zhuhai 519082, China

* Corresponding author: Yulin Zhao

Received  May 2019 Revised  February 2020 Published  May 2020

Fund Project: This research is supported by the NSF of China (No.11971495 and No.11801582)

This paper dealt with the existence of periodic waves for a perturbed quintic BBM equation by using geometric singular perturbation theory. By analyzing the perturbations of the Hamiltonian vector field with a hyperelliptic Hamiltonian of degree six, we proved that periodic wave solutions persist for sufficiently small perturbation parameter. It is also proved that the wave speed $ c_0(h) $ is decreasing on $ h $ by analyzing the ratio of Abelian integrals, where $ h $ is the energy level value. Moreover, the upper and lower bounds of the limit wave speed are given.

Citation: Lina Guo, Yulin Zhao. Existence of periodic waves for a perturbed quintic BBM equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 4689-4703. doi: 10.3934/dcds.2020198
References:
[1]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk., 18 (1963), 91-192.   Google Scholar

[2]

R. Asheghi and H. R. Z. Zangeneh, Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop, Comput. Math. Appl., 59 (2010), 1409-1418.  doi: 10.1016/j.camwa.2009.12.024.  Google Scholar

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T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.  Google Scholar

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R. Camassa and D. D. Holm, An integrable shallow wave equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[5]

A. Y. ChenL. N. Guo and X. J. Deng, Existence of solitary waves and periodic waves for a perturbed generalized BBM equation, J. Differential Equations, 261 (2016), 5324-5349.  doi: 10.1016/j.jde.2016.08.003.  Google Scholar

[6]

A. Y. ChenL. N. Guo and W. T. Huang, Existence of kink waves and periodic waves for a perturbed defocusing mKdV equation, Qual. Theory Dyn. Syst., 17 (2018), 495-517.  doi: 10.1007/s12346-017-0249-9.  Google Scholar

[7]

G. Derks and S. van Gils, On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations, Japan J. Indust. Appl. Math., 10 (1993), 413-430.  doi: 10.1007/BF03167282.  Google Scholar

[8]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[9]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid. Mech., 78 (1976), 237-246.  doi: 10.1017/S0022112076002425.  Google Scholar

[10]

C. K. R. T. Jones, Geometric singular perturbtion theory, in Dynamical Systems, Lecture Notes in Math., 1609, Springer, Berlin, 1995, 44–118. doi: 10.1007/BFb0095239.  Google Scholar

[11]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.   Google Scholar

[12]

D. J. Korteweg and G. de Vries, On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves, Philos. Mag. (5), 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[13]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20.   Google Scholar

[14]

T. Ogawa, Travelling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401-422.  doi: 10.32917/hmj/1206128032.  Google Scholar

[15]

T. Ogawa, Periodic travelling waves and their modulation, Japan J. Indust. Appl. Math., 18 (2001), 521-542.  doi: 10.1007/BF03168589.  Google Scholar

[16]

T. Ogawa and H. Suzuki, On the spectra of pulses in a nearly integrable system, SIAM J. Appl. Math., 57 (1997), 485-500.  doi: 10.1137/S0036139995288782.  Google Scholar

[17]

P. Rosenau, On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. Lett. A., 230 (1997), 305-318.  doi: 10.1016/S0375-9601(97)00241-7.  Google Scholar

[18]

J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous fluid, J. Phys. Soc. Japan, 44 (1978), 663-666.  doi: 10.1143/JPSJ.44.663.  Google Scholar

[19]

A. M. Wazwaz, Exact solution with compact and non-compact structures for the one-dimensional generalized Benjamin-Bona-Mahony equation, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 855-867.  doi: 10.1016/j.cnsns.2004.06.002.  Google Scholar

[20]

W. F. YanZ. R. Liu and Y. Liang, Existence of solitary waves and periodic waves to a perturbed generalized KdV equation, Math. Model. Anal., 19 (2014), 537-555.  doi: 10.3846/13926292.2014.960016.  Google Scholar

[21]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101, American Mathematical Society, Providence, RI, 1992.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk., 18 (1963), 91-192.   Google Scholar

[2]

R. Asheghi and H. R. Z. Zangeneh, Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop, Comput. Math. Appl., 59 (2010), 1409-1418.  doi: 10.1016/j.camwa.2009.12.024.  Google Scholar

[3]

T. B. BenjaminJ. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[4]

R. Camassa and D. D. Holm, An integrable shallow wave equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[5]

A. Y. ChenL. N. Guo and X. J. Deng, Existence of solitary waves and periodic waves for a perturbed generalized BBM equation, J. Differential Equations, 261 (2016), 5324-5349.  doi: 10.1016/j.jde.2016.08.003.  Google Scholar

[6]

A. Y. ChenL. N. Guo and W. T. Huang, Existence of kink waves and periodic waves for a perturbed defocusing mKdV equation, Qual. Theory Dyn. Syst., 17 (2018), 495-517.  doi: 10.1007/s12346-017-0249-9.  Google Scholar

[7]

G. Derks and S. van Gils, On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations, Japan J. Indust. Appl. Math., 10 (1993), 413-430.  doi: 10.1007/BF03167282.  Google Scholar

[8]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[9]

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, J. Fluid. Mech., 78 (1976), 237-246.  doi: 10.1017/S0022112076002425.  Google Scholar

[10]

C. K. R. T. Jones, Geometric singular perturbtion theory, in Dynamical Systems, Lecture Notes in Math., 1609, Springer, Berlin, 1995, 44–118. doi: 10.1007/BFb0095239.  Google Scholar

[11]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527-530.   Google Scholar

[12]

D. J. Korteweg and G. de Vries, On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves, Philos. Mag. (5), 39 (1895), 422-443.  doi: 10.1080/14786449508620739.  Google Scholar

[13]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20.   Google Scholar

[14]

T. Ogawa, Travelling wave solutions to a perturbed Korteweg-de Vries equation, Hiroshima Math. J., 24 (1994), 401-422.  doi: 10.32917/hmj/1206128032.  Google Scholar

[15]

T. Ogawa, Periodic travelling waves and their modulation, Japan J. Indust. Appl. Math., 18 (2001), 521-542.  doi: 10.1007/BF03168589.  Google Scholar

[16]

T. Ogawa and H. Suzuki, On the spectra of pulses in a nearly integrable system, SIAM J. Appl. Math., 57 (1997), 485-500.  doi: 10.1137/S0036139995288782.  Google Scholar

[17]

P. Rosenau, On nonanalytic solitary waves formed by a nonlinear dispersion, Phys. Lett. A., 230 (1997), 305-318.  doi: 10.1016/S0375-9601(97)00241-7.  Google Scholar

[18]

J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous fluid, J. Phys. Soc. Japan, 44 (1978), 663-666.  doi: 10.1143/JPSJ.44.663.  Google Scholar

[19]

A. M. Wazwaz, Exact solution with compact and non-compact structures for the one-dimensional generalized Benjamin-Bona-Mahony equation, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 855-867.  doi: 10.1016/j.cnsns.2004.06.002.  Google Scholar

[20]

W. F. YanZ. R. Liu and Y. Liang, Existence of solitary waves and periodic waves to a perturbed generalized KdV equation, Math. Model. Anal., 19 (2014), 537-555.  doi: 10.3846/13926292.2014.960016.  Google Scholar

[21]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101, American Mathematical Society, Providence, RI, 1992.  Google Scholar

Figure 1.  The phase portrait of system (1.13)
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