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:
[1]

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

[2]

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

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[37]

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

[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.

[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.

[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.

[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.

[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.

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

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[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.

[30]

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

[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.

[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.

[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.

[34]

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

[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.

[36]

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

[37]

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

[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.

[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.

[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.

[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.

[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.

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

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

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

Figure 1.  The portrait of system (12)
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