
-
Previous Article
Representation formula for symmetrical symplectic capacity and applications
- DCDS Home
- This Issue
-
Next Article
Fluctuations of ergodic sums on periodic orbits under specification
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.
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. |

[1] |
Xianbo Sun, Pei Yu. Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 965-987. doi: 10.3934/dcdsb.2018341 |
[2] |
Melek Jellouli. On the controllability of the BBM equation. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022002 |
[3] |
Amin Esfahani. Remarks on a two dimensional BBM type equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1111-1127. doi: 10.3934/cpaa.2012.11.1111 |
[4] |
Mahendra Panthee. On the ill-posedness result for the BBM equation. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 253-259. doi: 10.3934/dcds.2011.30.253 |
[5] |
Xavier Carvajal, Mahendra Panthee. On ill-posedness for the generalized BBM equation. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4565-4576. doi: 10.3934/dcds.2014.34.4565 |
[6] |
Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241 |
[7] |
Yvan Martel, Frank Merle. Inelastic interaction of nearly equal solitons for the BBM equation. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 487-532. doi: 10.3934/dcds.2010.27.487 |
[8] |
Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345 |
[9] |
Jean-Paul Chehab, Pierre Garnier, Youcef Mammeri. Long-time behavior of solutions of a BBM equation with generalized damping. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1897-1915. doi: 10.3934/dcdsb.2015.20.1897 |
[10] |
Khaled El Dika. Smoothing effect of the generalized BBM equation for localized solutions moving to the right. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 973-982. doi: 10.3934/dcds.2005.12.973 |
[11] |
Jerry L. Bona, Hongqiu Chen, Chun-Hsiung Hsia. Well-posedness for the BBM-equation in a quarter plane. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1149-1163. doi: 10.3934/dcdss.2014.7.1149 |
[12] |
Zengji Du, Xiaojie Lin, Yulin Ren. Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping. Communications on Pure and Applied Analysis, 2022, 21 (6) : 1987-2003. doi: 10.3934/cpaa.2021118 |
[13] |
Jaime Angulo Pava, César A. Hernández Melo. On stability properties of the Cubic-Quintic Schródinger equation with $\delta$-point interaction. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2093-2116. doi: 10.3934/cpaa.2019094 |
[14] |
Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks and Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555 |
[15] |
Wenxiong Chen, Congming Li, Biao Ou. Qualitative properties of solutions for an integral equation. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 347-354. doi: 10.3934/dcds.2005.12.347 |
[16] |
Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878 |
[17] |
Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic and Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261 |
[18] |
Jerry Bona, Hongqiu Chen, Shu Ming Sun, B.-Y. Zhang. Comparison of quarter-plane and two-point boundary value problems: the BBM-equation. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 921-940. doi: 10.3934/dcds.2005.13.921 |
[19] |
Justin Forlano. Almost sure global well posedness for the BBM equation with infinite $ L^{2} $ initial data. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 267-318. doi: 10.3934/dcds.2020011 |
[20] |
Ming Wang. Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5763-5788. doi: 10.3934/dcds.2016053 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]