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 and Continuous Dynamical Systems, 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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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. 

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

Figure 1.  The phase portrait of system (1.13)
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