February  2019, 24(2): 965-987. doi: 10.3934/dcdsb.2018341

Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms

Department of Applied Mathematics, Western University, London, Ontario, N6A 5B7, Canada

* Corresponding author: Pei Yu

Received  May 2017 Revised  September 2017 Published  November 2018

In this paper, we consider a generalized BBM equation with weak backward diffusion, dissipation and Marangoni effects, and study the existence of periodic and solitary waves. Main attention is focused on periodic and solitary waves on a manifold via studying the number of zeros of some linear combination of Abelian integrals. The uniqueness of the periodic waves is established when the equation contains one coefficient in backward diffusion and dissipation terms, by showing that the Abelian integrals form a Chebyshev set. The monotonicity of the wave speed is proved, and moreover the upper and lower bounds of the limiting wave speeds are obtained. Especially, when the equation involves Marangoni effect due to imposed weak thermal gradients, it is shown that at most two periodic waves can exist. The exact conditions are obtained for the existence of one and two periodic waves as well as for the co-existence of one solitary and one periodic waves. The analysis is mainly based on Chebyshev criteria and asymptotic expansions of Abelian integrals near the solitary and singularity.

Citation: Xianbo Sun, Pei Yu. Periodic traveling waves in a generalized BBM equation with weak backward diffusion and dissipation terms. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 965-987. doi: 10.3934/dcdsb.2018341
References:
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A. ChenL. Guo and X. 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

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G. Derks and S. 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

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C. ElphickG. R. IerleyO. Regev and E. A. Spiegel, Interacting localized structures with Galilean invariance, Phys. Rev. A, 44 (1991), 1110-1122.   Google Scholar

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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

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P. L. Garcia-YbarraJ. L. Castillo and M. G. Velarde, Bénard-Marangoni convection with a deformable interface and poorly conducting boundaries, Phys. Fluids, 30 (1987), 2655-2661.   Google Scholar

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A. Geyera and J. Villadelpratb, On the wave length of smooth periodic traveling waves of the Camassa-Holm equation, J. Differential Equations, 259 (2015), 2317-2332.  doi: 10.1016/j.jde.2015.03.027.  Google Scholar

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M. GrauF. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.  Google Scholar

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W. Hai and Y. Xiao, Soliton solution of a singularly perturbed KdV equation, Phys. Lett. A, 208 (1995), 79-83.  doi: 10.1016/0375-9601(95)00729-M.  Google Scholar

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J. M. Hyman and B. Nicolaenko, Coherence and chaos in Kuramoto-Velarde equation, Directions in Partial Differential Equations (Madison, WI, 1985), 89-111, Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Boston, MA, 1987.  Google Scholar

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E. R. Johnsona and D. E. Pelinovsky, Orbital stability of periodic waves in the class of reduced Ostrovsky equations, J. Differential Equations, 261 (2016), 3268-3304.  doi: 10.1016/j.jde.2016.05.026.  Google Scholar

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T. Kawahara and S. Toh, Pulse interactions in an unstable dissipative-dispersion nonlinear system, Phys. Fluids, 31 (1988), 2103-2111.  doi: 10.1063/1.866610.  Google Scholar

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A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530.   Google Scholar

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

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N. A. Kudryashev, Exact soliton solutions for a generalized equation of evolution for the wave dynamics, Prikl. Math. Mekh., 52 (1988), 465-470.  doi: 10.1016/0021-8928(88)90090-1.  Google Scholar

[28]

S. Y. LouG. X. Huang and X.-Y. Ruan, Exact solitary waves in a convecting fluid, J. Phys. A: Math. Gen., 24 (1991), 587-590.  doi: 10.1088/0305-4470/24/11/003.  Google Scholar

[29]

F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differential Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.  Google Scholar

[30]

M. B. A. Mansour, Travelling wave solutions for a singularly perturbed Burgers-KdV equation, Pramana J. Phys., 73 (2009), 799-806.   Google Scholar

[31]

M. B. A. Mansour, Traveling waves for a dissipative modified KdV equation, J. Egypt. Math. Soc., 20 (2012), 134-138.  doi: 10.1016/j.joems.2012.08.002.  Google Scholar

[32]

M. B. A. Mansour, A geometric construction of traveling waves in a generalized nonlinear dispersive-dissipative equation, J. Geom. Phy., 69 (2013), 116-122.  doi: 10.1016/j.geomphys.2013.03.004.  Google Scholar

[33]

S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation, SIAM J. Control. Optim., 39 (2011), 1677-1696.  doi: 10.1137/S0363012999362499.  Google Scholar

[34]

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

[35]

C. NormadY. Pomeau and M. G. Velarde, Convective instability: A physicist's approach, Rev. Mod. Phys., 49 (1977), 581-624.  doi: 10.1103/RevModPhys.49.581.  Google Scholar

[36]

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

[37]

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

[38]

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

[39]

K. Omrani, The convergence of fully discrete Galerkin approximations for the Benjamin-Bona-Mahony(BBM)equation, Appl. Math. Comput., 180 (2006), 614-621.  doi: 10.1016/j.amc.2005.12.046.  Google Scholar

[40]

Y. PomeauA. Ramani and B. Grammaticos, Structural stability of the Korteweg-de Vries solitons under a singular perturbation, Physica D, 31 (1988), 127-134.  doi: 10.1016/0167-2789(88)90018-8.  Google Scholar

[41]

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

[42]

J. Tyson and J. Keener, Singular perturbation theory of traveling waves in excitable media, Physica D, 32 (1988), 327-361.  doi: 10.1016/0167-2789(88)90062-0.  Google Scholar

[43] M. G. Velarde, Physicochemical Hydrodynamics: Interfacial Phenomena, Plenum, New York, 1987.   Google Scholar
[44]

M. G. Velarde and C. Normand, Natural convection, Sci. Am., 243 (1980), 92-106.   Google Scholar

[45] J. Von Zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, 2013.  doi: 10.1017/CBO9781139856065.  Google Scholar
[46]

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

[47]

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

[48]

J. Yang, A normal form for Hamiltonian-Hopf bifurcations in nonlinear Schrodinger equations with general external potentials, SIAM J. Appl. Math., 76 (2016), 598-617.  doi: 10.1137/15M1042619.  Google Scholar

[49]

W. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differential Equations, 155 (1988), 89-132.  doi: 10.1006/jdeq.1998.3584.  Google Scholar

[50]

H. ZangM. Han and D. Xiao, On Abelian integrals of a homoclinic loop through a nilpotent saddle for planar near-Hamiltonian systems, J. Differential Equations, 245 (2008), 1086-1111.  doi: 10.1016/j.jde.2008.04.018.  Google Scholar

[51]

X. ZhaoW. XuS. Li and J. Shen, Bifurcations of traveling wave solutions for a class of the generalized Benjamin-Bona-Mahony equation, Appl. Math. Comput., 175 (2006), 1760-1774.  doi: 10.1016/j.amc.2005.09.019.  Google Scholar

[52]

Y. ZhouQ. Liu and W. Zhang, Bounded traveling waves of the Burgers-Huxley equation, Nonlinear Analysis (TMA), 74 (2011), 1047-1060.  doi: 10.1016/j.na.2010.09.012.  Google Scholar

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Y. Zhou, Q. Liu and W. Zhang, Bounded traveling wave of the generalized Burgers-Fisher equation, Int. J. Bifur. Chaos, 23 (2013), 1350054 (11 pages). doi: 10.1142/S0218127413500545.  Google Scholar

show all references

References:
[1]

B. Amitabha, A geometric approach to singularly perturbed nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 431-454.   Google Scholar

[2]

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

[3]

E. BenilovR. Grimshaw and E. Kuznetsov, The generation of radiating waves in a singularly-perturbed Korteweg-de Vries equation, Physica D, 69 (1993), 270-278.  doi: 10.1016/0167-2789(93)90091-E.  Google Scholar

[4]

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

[5]

F. H. Busse, Non-linear properties of thermal convection, Reports on Progress in Physics, 41 (1978), 1929-1967.   Google Scholar

[6]

R. Camassa and 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

[7]

A. ChenL. Guo and X. 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

[8]

C. I. Christov and M. G. Velarde, Dissipative solitons, Physica D, 86 (1995), 323-347.  doi: 10.1016/0167-2789(95)00111-G.  Google Scholar

[9]

G. Derks and S. 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

[10]

C. ElphickG. R. IerleyO. Regev and E. A. Spiegel, Interacting localized structures with Galilean invariance, Phys. Rev. A, 44 (1991), 1110-1122.   Google Scholar

[11]

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

[12]

P. L. Garcia-YbarraJ. L. Castillo and M. G. Velarde, Bénard-Marangoni convection with a deformable interface and poorly conducting boundaries, Phys. Fluids, 30 (1987), 2655-2661.   Google Scholar

[13]

A. Geyera and J. Villadelpratb, On the wave length of smooth periodic traveling waves of the Camassa-Holm equation, J. Differential Equations, 259 (2015), 2317-2332.  doi: 10.1016/j.jde.2015.03.027.  Google Scholar

[14]

M. GrauF. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.  Google Scholar

[15]

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.   Google Scholar

[16]

W. Hai and Y. Xiao, Soliton solution of a singularly perturbed KdV equation, Phys. Lett. A, 208 (1995), 79-83.  doi: 10.1016/0375-9601(95)00729-M.  Google Scholar

[17]

S. HakkaevI. D. Iliev and K. Kirchev, Stability of periodic travelling shallow-water waves determined by Newton's equation, J. Phys. A, 41 (2008), 085203, 31pp.  doi: 10.1088/1751-8113/41/8/085203.  Google Scholar

[18]

M. Han, Asymptotic expansions of Abelian integrals and limit cycle bifurcations, Int. J. Bifur. Chaos, 22 (2012), 1250296 (30 pages). doi: 10.1142/S0218127412502963.  Google Scholar

[19]

M. Han, J. Yang and D. Xiao, Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle, Int. J. Bifur. Chaos 22 (2012), 1250189 (33 pages). doi: 10.1142/S0218127412501891.  Google Scholar

[20]

J. M. Hyman and B. Nicolaenko, The Kuramoto-Sivashinsky equation: A bridge between PDE's and dynamical systems, Physica D, 18 (1986), 113-126.  doi: 10.1016/0167-2789(86)90166-1.  Google Scholar

[21]

J. M. Hyman and B. Nicolaenko, Coherence and chaos in Kuramoto-Velarde equation, Directions in Partial Differential Equations (Madison, WI, 1985), 89-111, Publ. Math. Res. Center Univ. Wisconsin, 54, Academic Press, Boston, MA, 1987.  Google Scholar

[22]

E. R. Johnsona and D. E. Pelinovsky, Orbital stability of periodic waves in the class of reduced Ostrovsky equations, J. Differential Equations, 261 (2016), 3268-3304.  doi: 10.1016/j.jde.2016.05.026.  Google Scholar

[23] V. I. Karpman, Non-Linear Waves in Dispersive Media, Pergamon Press, Oxford-New York-Toronto, Ont., 1975.   Google Scholar
[24]

T. Kawahara and S. Toh, Pulse interactions in an unstable dissipative-dispersion nonlinear system, Phys. Fluids, 31 (1988), 2103-2111.  doi: 10.1063/1.866610.  Google Scholar

[25]

A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530.   Google Scholar

[26]

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

[27]

N. A. Kudryashev, Exact soliton solutions for a generalized equation of evolution for the wave dynamics, Prikl. Math. Mekh., 52 (1988), 465-470.  doi: 10.1016/0021-8928(88)90090-1.  Google Scholar

[28]

S. Y. LouG. X. Huang and X.-Y. Ruan, Exact solitary waves in a convecting fluid, J. Phys. A: Math. Gen., 24 (1991), 587-590.  doi: 10.1088/0305-4470/24/11/003.  Google Scholar

[29]

F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differential Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.  Google Scholar

[30]

M. B. A. Mansour, Travelling wave solutions for a singularly perturbed Burgers-KdV equation, Pramana J. Phys., 73 (2009), 799-806.   Google Scholar

[31]

M. B. A. Mansour, Traveling waves for a dissipative modified KdV equation, J. Egypt. Math. Soc., 20 (2012), 134-138.  doi: 10.1016/j.joems.2012.08.002.  Google Scholar

[32]

M. B. A. Mansour, A geometric construction of traveling waves in a generalized nonlinear dispersive-dissipative equation, J. Geom. Phy., 69 (2013), 116-122.  doi: 10.1016/j.geomphys.2013.03.004.  Google Scholar

[33]

S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation, SIAM J. Control. Optim., 39 (2011), 1677-1696.  doi: 10.1137/S0363012999362499.  Google Scholar

[34]

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

[35]

C. NormadY. Pomeau and M. G. Velarde, Convective instability: A physicist's approach, Rev. Mod. Phys., 49 (1977), 581-624.  doi: 10.1103/RevModPhys.49.581.  Google Scholar

[36]

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

[37]

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

[38]

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

[39]

K. Omrani, The convergence of fully discrete Galerkin approximations for the Benjamin-Bona-Mahony(BBM)equation, Appl. Math. Comput., 180 (2006), 614-621.  doi: 10.1016/j.amc.2005.12.046.  Google Scholar

[40]

Y. PomeauA. Ramani and B. Grammaticos, Structural stability of the Korteweg-de Vries solitons under a singular perturbation, Physica D, 31 (1988), 127-134.  doi: 10.1016/0167-2789(88)90018-8.  Google Scholar

[41]

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

[42]

J. Tyson and J. Keener, Singular perturbation theory of traveling waves in excitable media, Physica D, 32 (1988), 327-361.  doi: 10.1016/0167-2789(88)90062-0.  Google Scholar

[43] M. G. Velarde, Physicochemical Hydrodynamics: Interfacial Phenomena, Plenum, New York, 1987.   Google Scholar
[44]

M. G. Velarde and C. Normand, Natural convection, Sci. Am., 243 (1980), 92-106.   Google Scholar

[45] J. Von Zur Gathen and J. Gerhard, Modern Computer Algebra, Cambridge University Press, 2013.  doi: 10.1017/CBO9781139856065.  Google Scholar
[46]

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

[47]

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

[48]

J. Yang, A normal form for Hamiltonian-Hopf bifurcations in nonlinear Schrodinger equations with general external potentials, SIAM J. Appl. Math., 76 (2016), 598-617.  doi: 10.1137/15M1042619.  Google Scholar

[49]

W. Yong, Singular perturbations of first-order hyperbolic systems with stiff source terms, J. Differential Equations, 155 (1988), 89-132.  doi: 10.1006/jdeq.1998.3584.  Google Scholar

[50]

H. ZangM. Han and D. Xiao, On Abelian integrals of a homoclinic loop through a nilpotent saddle for planar near-Hamiltonian systems, J. Differential Equations, 245 (2008), 1086-1111.  doi: 10.1016/j.jde.2008.04.018.  Google Scholar

[51]

X. ZhaoW. XuS. Li and J. Shen, Bifurcations of traveling wave solutions for a class of the generalized Benjamin-Bona-Mahony equation, Appl. Math. Comput., 175 (2006), 1760-1774.  doi: 10.1016/j.amc.2005.09.019.  Google Scholar

[52]

Y. ZhouQ. Liu and W. Zhang, Bounded traveling waves of the Burgers-Huxley equation, Nonlinear Analysis (TMA), 74 (2011), 1047-1060.  doi: 10.1016/j.na.2010.09.012.  Google Scholar

[53]

Y. Zhou, Q. Liu and W. Zhang, Bounded traveling wave of the generalized Burgers-Fisher equation, Int. J. Bifur. Chaos, 23 (2013), 1350054 (11 pages). doi: 10.1142/S0218127413500545.  Google Scholar

Figure 1.  The portrait of system (12)
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