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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 |
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.
Mathematics Subject Classification:
Primary: 34C25, 34C60; Secondary: 37C27.
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
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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. Benjamin, J. 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. Chen, L. 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. Chen, L. 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. Yan, Z. 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. Benjamin, J. 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. Chen, L. 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. Chen, L. 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. Yan, Z. 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. |
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